Section 6.1: More About Areas
1. Find the area of the region bounded by the curves and
a. e.
b. f.
c. g.
d. h.
ANS: A
2. Find the area of the region bounded by the curves and
a. e.
b. f.
c. g.
d. h.
ANS: G
3. The area of the region bounded by and between and is
a. e.
b. f.
c. g.
d. 0 h.
ANS: A
4. Find the area of the region bounded by the curves and
a. e.
b. f.
c. g. 4
d. h. 2
ANS: E
5. Find the area of the region bounded by the curves and
a. e.
b. f.
c. g.
d. 20 h.
ANS: F
6. The area of the region bounded by and the y-axis is
a. e.
b. f. 1
c. g. 2
d. 0 h.
ANS: B
7. Find the area of the region bounded by the parabola and the line .
a. e.
b. f.
c. g.
d. h.
ANS: B
8. Find the area of the region bounded by the curve , and the x-axis.
a. e.
b. f.
c. g.
d. h.
ANS: D
9. Find the area of the region bounded by , and the x-axis.
a. e.
b. f.
c. g.
d. h.
ANS: E
10. Find the area of the region bounded by , and the x-axis.
a. e.
b. f.
c. g.
d. h.
ANS: C
11. Find the area of the region bounded by the curves and .
ANS:
12. Find the area of the region bounded by the curves and .
ANS:
13. Find the area of the region bounded by the curves and the x-axis.
ANS:
14. Find the area of the region bounded by the curves and .
ANS:
15. Find the area of the region bounded by the curves and .
ANS:
16. Let R be the region bounded by: , the tangent to at , and the x-axis.
Find the area of R integrating
(a) with respect to x.
(b) with respect to y.
ANS:
(a)
(b)
17. Find the area of the region bounded by the curves and .
ANS:
About 10.43
18. Using the help of a graphing calculator, find the area of the region bounded by the curves and .
ANS:
19. Find the area of the region bounded by the curves
ANS:
20. Find the area of the shaded region:
ANS:
21. Find the area of the shaded region:
ANS:
22. Find the area of the shaded region:
ANS:
23. Find the area of the shaded region:
ANS:
9
24. Find the area of the region bounded by
ANS:
25. Find the area of the region bounded by
ANS:
8
26. A particle is moving in a straight line and its velocity is given by where t is measure in seconds and v in meters per second. Find the distance traveled by the particle during the time interval .
ANS:
28 m
27. A stone is thrown straight up from the top of a tower that is 80 ft tall with initial velocity 64 ft/s. What is the total distance traveled by the stone when it hits the ground?
ANS:
208 feet
28. Express the area of the given region as a definite integral. Do not evaluate.
ANS:
OR
29. Express the area of the given region as a definite integral. Do not evaluate.
ANS:
OR
Section 6.2: Volumes
1. Find the volume of the solid obtained when the region bounded by the x-axis, the y-axis, and the line is rotated about the x-axis.
a. e.
b. f.
c. g.
d. h.
ANS: E
2. Find the volume of the solid obtained when the region bounded by the line , the line , and the x-axis is rotated about the y-axis.
a. e.
b. f.
c. g.
d. h.
ANS: A
3. Find the volume of the solid obtained when the region bounded by the curve and the x-axis is rotated about the x-axis.
a. e.
b. f.
c. g.
d. h.
ANS: A
4. The base of a solid is the parabolic region . Cross-sections perpendicular to the x-axis are squares. Find the volume of .
a. 1.5 e. 1.9
b. 1.6 f. 2.0
c. 1.7 g. 2.1
d. 1.8 h. 2.2
ANS: F
5. Find the volume of the solid obtained by rotating about the line , the region bounded by .
a. e.
b. f.
c. g.
d. h.
ANS: F
6. The volume of the solid obtained by rotating the region about the x-axis is
a. e.
b. f.
c. g.
d. h.
ANS: B
7. A solid has a circular base of radius 1. Parallel cross-sections perpendicular to the base are equilateral triangles. Find the volume of the solid.
a. e.
b. f.
c. g.
d. h.
ANS: F
8. Find the volume of the solid obtained when the region bounded by the curves is rotated about the line x =1.
a. e.
b. f.
c. g.
d. h.
ANS: F
9. Find the volume of the solid obtained when the region bounded by the curves , , and the x-axis is rotated about the line x = 1.
a. e.
b. f.
c. g.
d. h.
ANS: G
10. Find the volume of the solid obtained by rotating the region bounded by the curves about the x-axis.
ANS:
11. Find the volume of the solid obtained by rotating the region bounded by the curves about the y-axis.
ANS:
12. Consider the region in the xy-plane between and . Find the volume of the solid generated by rotating this region about the x-axis.
ANS:
13. Find the volume of the solid formed when the region bounded by the curves is rotated about the x-axis.
ANS:
14. Find the volume of the solid generated by rotating about the line the region bounded by the graphs of the equations
ANS:
15. Find the volume of the solid obtained by rotating the region bounded by the curves about the line y = 2.
ANS:
16. Find the volume of the solid obtained by rotating the region bounded by the curves about the x-axis.
ANS:
17. Find the volume of the solid obtained by rotating the region bounded by the curves about the line x = 2.
ANS:
18. Find the volume of the solid obtained by rotating the region bounded by the curve and the x-axis about the line y = 2.
ANS:
19. Find the volume of the solid generated by rotating the region bounded by and the x-axis about the x-axis.
ANS:
20. The base of a certain solid is a plane region R enclosed by the x-axis and the curve . Each cross-section of the solid perpendicular to the y-axis is an isosceles triangle of height 4 with its base lying in R. Find the volume of the solid.
ANS:
21. The base of a certain solid is a plane region R enclosed by the x-axis and the curve . Each cross-section of the solid perpendicular to the y-axis is an equilateral triangle with its base lying in R. Find the volume of the solid.
ANS:
22. The base of a certain solid is a plane region R enclosed by the x-axis and the curve . Each cross-section of the solid perpendicular to the y-axis is an isosceles right triangle with hypotenuse lying in R. Find the volume of the solid.
ANS:
23. The base of a certain solid is the triangular region with vertices (0,0), (1,1), and (2,0). Cross-sections perpendicular to the x-axis are semicircles. Find the volume of the solid.
ANS:
24. The base of a certain solid is an elliptical region with boundary curve Cross-sections perpendicular to the x-axis are squares. Find the volume of the solid.
ANS:
192
Section 6.3: Volumes by Cylindrical Shells
1. The volume of the solid obtained by rotating the plane region enclosed by about the y-axis is
a. e.
b. f.
c. g.
d. 0 h. 1
ANS: B
2. The volume of the solid obtained by rotating the plane region enclosed by about the x = 1 is
a. e.
b. f.
c. g.
d. h.
ANS: C
3. Find the volume of the solid obtained when the region bounded by the curve and the x-axis is rotated about the y-axis.
a. e.
b. f.
c. g.
d. h.
ANS: E
4. Find the volume of the solid obtained when the region above the x-axis, bounded by the x-axis and the curve , is rotated about the y-axis.
a. e.
b. f.
c. g.
d. h.
ANS: C
5. The volume of the solid obtained by rotating the plane region enclosed by the y-axis and the x-axis about the y-axis is
a. e.
b. f.
c. 2 g. 4
d. 1 h.
ANS: A
6. Find the volume of the solid obtained by rotating the region bounded by xy = 1, y = 1, y = 2 and the y-axis about the x-axis.
ANS:
7. Find the volume of the solid obtained by rotating the region bounded by the y-axis and the x-axis about the y-axis.
ANS:
8. Find the volume of the solid obtained by rotating the region bounded by and the x-axis about the line x = 2.
ANS:
9. Find the volume of the solid obtained by rotating the region bounded by and x = 1 about the line x = 1. (Use a graphing calculator.)
ANS:
10. Let R be region bounded by the graph of , and the line y = 3.
(a) Find the volume of the solid obtained by rotating R about the x-axis.
(b) Find the volume of the solid obtained by rotating R about the y-axis.
(c) Find the volume of the solid obtained by rotating R about the line y = 3.
(d) Find the volume of the solid obtained by rotating R about the line x = 2.
ANS:
(a)
(b)
(c)
(d)
11. Let R be region bounded by the curve .
(a) Find the volume of the solid obtained by rotating R about the x-axis.
(b) Find the volume of the solid obtained by rotating R about the line x = 5.
(c) Find the volume of the solid obtained by rotating R about the line y = –1.
ANS:
(a)
(b) 72
(c)
12. The region R is given by the shaded area in the figure below:
(a) Find the area of the shaded region R.
(b) Find the volume of the solid obtained by rotating R about
(i) the x-axis. (ii) the y-axis. (iii) the line x = 2. (iv) the line y = 4
ANS:
(a)
(b) (i) (ii) (iii) (iv)
13. A hole of radius 6 cm is drilled through the center of a sphere of radius 10 cm. How much of the ball’s volume is removed?
ANS:
Section 6.4: Arc Length
1. Find the arc length of the curve from (3, 4) to (9, 12).
a. 13 e. 9
b. 10 f. 15
c. 8 g. 11
d. 14 h. 12
ANS: B
2. Find the arc length of the curve from (0, 0) to ( , ).
a. e.
b. f.
c. g.
d. h.
ANS: E
3. Find the arc length of the curve .
a. e.
b. f.
c. g.
d. h.
ANS: A
4. Find the arc length of the curve .
a. e.
b. f.
c. g.
d. h.
ANS: F
5. Find the length of the curve .
a. 19 e.
b. f.
c. g. 38
d. 55 h. 65
ANS: F
6. Find the length of the curve .
a. 81 e. 57
b. f.
c. g.
d. h.
ANS: E
7. Find the length of the curve .
a. 0 e. 12
b. f. 6
c. 2 g. 4
d. h. 2
ANS: F
8. Find the length of the curve .
a. 0 e. 1
b. 8 f. 6
c. 2 g. 4
d. 12 h. 9
ANS: D
9. Give a definite integral representing the length of the parametric curve .
a. e.
b. f.
c. g.
d. h.
ANS: E
10. Find the length of the curve
a. 0 e. 1
b. f.
c. 2 g. 4
d. h.
ANS: H
11. Give a definite integral representing the length of the curve .
a. e.
b. f.
c. g.
d. h.
ANS: B
12. Find the length of the curve
ANS:
13. Find the length of the curve using a graphing calculator to evaluate the integral
ANS:
14. Find the length of
ANS:
15. Set up, but do not evaluate, an integral for the length of
ANS:
16. Set up, but do not evaluate, an integral for the length of
ANS:
17. Find the length of the curve .
ANS:
18. Find the distance traveled by a particle with position as t varies in the time interval . Compare with the length of the curve.
ANS:
distance =
19. If represents the position of a particle, find the distance the particle travels as t moves from 0 to
ANS:
20. Find the length of the curve using a graphing calculator to evaluate the integral.
ANS:
21. The equation of a curve in parametric form is Find the arc length of the curve from .
ANS:
22. A curve is written parametrically as Find the arc length of the curve from .
ANS:
4
23. Find the length of the curve
ANS:
24. Find the arc length of the curve .
ANS:
25. Find the arc length of the curve .
ANS:
Section 6.5: Average Value of a Function
1. Find the average value of the function .
a. 0 e.
b. f. 5
c. 10 g. 4
d. h.
ANS: E
2. The density of a rod 9 meters long is kg/m at a distance of x meters from one end of the rod. Find the average density of the rod.
a. 6 e. 2
b. f.
c. 3 g. 4
d. 1 h.
ANS: E
3. Find the average value of the function
a. 1 e.
b. f.
c. 2 g. 0
d. 6 h.
ANS: F
4. Find the average value of the function on the interval .
a. e. 2
b. f. 4
c. 8 g. 16
d. 3 h. 6
ANS: A
5. Let , find c such that on the interval .
ANS:
6. Let , find c such that on the interval .
ANS:
7. Find the average value of the function whose graph is given below.
ANS:
0
8. Find the average value of the function whose graph is given below.
ANS:
9. Estimate the average value of the function whose graph is given below.
ANS:
2
10. Find the average value of on the interval . At how many points in the interval does have this value?
ANS:
11. The temperature (in °F) in a certain city t hours after 9 A.M. is approximated by the function . Find the average temperature during the period from 9 A.M. to 9 P.M..
ANS:
59° F
12. The temperature (in °C) of a metal rod 5 m long is 4x at a distance x meters from one end of the rod. What is the average temperature of the rod?
ANS:
10° C
13. A stone is dropped from a bell tower 100 feet tall. Find the average velocity of the stone from the instant it is dropped until it strikes the ground. (Assume that the acceleration due to gravity is 32 ft/s2.)
ANS:
40 ft/s
14. A culture of bacteria is doubling every hour. What is the average population over the first two hours if we assume that the culture initially contained two million organisms?
ANS:
million 4,328,085
15. Find the average value of on the interval .
ANS:
5
16. A particle is moving along a straight line so that its velocity at time t is v(t) = 3t2. At what time t during the interval is its velocity the same as its average velocity over the entire interval?
ANS:
17. The following table shows the velocity of a car (in mi/hr) during the first five seconds of a race.
t (s) 0 1 2 3 4 5
v (mi/h ) 0 20 32 46 54 62
Determine the average velocity of the car during this five-second interval.
ANS:
Answer may vary between 30.4 mph and 42.8 mph.
18. The graph of a continuous function g(x) is given below:
List from smallest to largest:
(a) The average value of g over (d)
(b) The average rate of change of g over (e)
(c) (f)
ANS:
(f) < (d) < (b) < (a) < (c) < (e)
19. Consider the region R bounded by , and the y-axis.
(a) Find the area of R.
(b) Find the average height of R.
(c) Find the volume, V, of the solid obtained by rotating R about the x-axis.
(d) A cross section of the solid generated by part (c) taken perpendicular to the x-axis is a washer. Determine the average area of the cross sections of the solid.
ANS:
(a)
(b)
(c) 2
(d)
20. The voltage (in volts) at an electrical outlet is a function of time (in seconds) given by V(t) = V0 cos(120 ) where V0 is a constant representing the maximum voltage.
(a) What is the average value of the voltage over one second?
(b) How many times does the voltage reach a maximum in one second?
(c) Define the new function . Compute , the average value of over one cycle.
(d) Instead of the average voltage, engineers use the root mean square . Determine Vrms in terms of V0.
(e) The standard household voltage in the United Stated is 100 volts. This means that Vrms = 110. What is the value of V0?
ANS:
(a) 0
(b) 61 for
(c)
(d) Vrms V0
(e) V0
Section 6.6: Application to Physics and Engineering
1. A force of 20 pounds stretches a spring foot beyond its natural length. Find the work done in stretching the spring 1 foot.
a. 20 ft-lb e. ft-lb
b. ft-lb f. ft-lb
c. ft-lb g. ft-lb
d. ft-lb h. ft-lb
ANS: D
2. A spring stretches 1 foot beyond its natural position under a force of 100 points. How much work in foot-pounds is done in stretching it 3 feet beyond its natural position?
a. 600 e. 100
b. 30 f. 900
c. 1500 g. 150
d. 450 h. 300
ANS: D
3. A rope 100 feet long weighing 2 pounds per foot hangs over the edge of a building 100 feet tall. How much work in foot-pounds is done in pulling the rope to the top of the building?
a. 7500 e. 10,000
b. 5000 f. 750
c. 1000 g. 1250
d. 500 h. 12,500
ANS: E
4. Find the work (in ft-lb) done in raising 500 lb of ore from a mine that is 1000 ft deep. Assume that the cable used to raise the ore weighs 2 lb/ft.
a. e. 106
b. f.
c. g.
d. h.
ANS: F
5. A right circular cylinder tank of height 1 foot and radius 1 foot is full of water. Taking the density of water to be a nice round 60 pounds per cubic foot, how much work in foot-pounds is required to pump all of the water up and over the top of the tank?
a. 8 e. 5
b. 24 f. 4
c. 16 g. 30
d. 6 h. 18
ANS: G
6. Find the center of mass of the linear system .
a. e.
b. f.
c. 6 g.
d. h.
ANS: A
7. Find the center of mass of the linear system
a. 9 e.
b. f.
c. g.
d. 0 h. 18
ANS: B
8. Find the center of mass of the system
a. e.
b. f.
c. g.
d. h.
ANS: A
9. Find the x-coordinate of the centroid of the region bounded by the graphs and the x-axis.
a. 8 e. 4
b. f.
c. g.
d. h.
ANS: F
10. Find the y-coordinate of the centroid of the region bounded by the curves
a. 0.85 e. 0.75
b. 0.70 f. 0.55
c. 0.60 g. 0.80
d. 0.50 h. 0.65
ANS: C
11. Find the x-coordinate x at the centroid of the region bounded by the x-axis and the lines
a. e.
b. f.
c. g.
d. h.
ANS: A
12. An aquarium 1 foot high, 1 foot wide, and 2 feet long is filled with water. For simplicity, take the density of water to be 60 lb/ft3 . Find the hydrostatic force in pound on one of the 1 foot by 2 foot sides of the aquarium.
a. 336 e. 60
b. 30 f. 120
c. 168 g. 28
d. 240 h. 12
ANS: E
13. A gate in an irrigation canal is in the form of a trapezoid 3 feet wide at the bottom, 5 feet wide at the top, with height equal to 2 feet. It is placed vertically in the canal with the water extending to its top. For simplicity, take the density of water to be 60 lb/ft3. Find the hydrostatic force in pounds on the gate.
a. 360 e. 400
b. 380 f. 460
c. 440 g. 500
d. 420 h. 480
ANS: C
14. A right circular cylinder tank of height 1 foot and radius 1 foot is full of water. Taking the density of water to be a nice round 60 pounds per cubic foot, find the hydrostatic force in pounds on the side of the tank.
a. 30 e. 30
b. 60 f. 120
c. 240 g. 60
d. 240 h. 120
ANS: G
15. A swimming pool 24 feet long and 15 feet wide has a bottom that is an inclined plane, the shallow end having a depth of 3 feet, and the deep end 10 feet. The pool is filled with water. For simplicity, take the density of water to be 60 lb/ft3. Find the hydrostatic force in pounds on the bottom of the pool.
a. 145,750 e. 146,500
b. 146,250 f. 147,500
c. 147,000 g. 146,750
d. 147,250 h. 146,000
ANS: B
16. If ft-lb of work is needed to stretch a spring a length ft beyond its natural length, find the spring constant.
ANS:
30 lb/ft
17. A force of 8 dynes is required to stretch a spring from its natural length of 10 cm to a length of 15 cm. How much work is done
(a) in stretching the spring to a length of 25 cm?
(b) in stretching the spring from a length of 20 cm to a length of 25 cm?
ANS:
(a) 180 dynes-cm
(b) 100 dynes-cm
18. A force of 10 pounds is required to stretch a spring from its natural length of 8 inches to a length of 10 inches. How much work is done in stretching the spring to a length of 12 inches?
ANS:
40 lb-inches
19. How much work is done to bring a load of ore to the surface if a miner uses a cable weighing 2 lb/ft to haul a 100 lb bucket of ore up a mine shaft 800 feet deep?
ANS:
720,000 ft-lb
20. A 100-foot length of steel chain weighing 5 lb/ft is dangling from the drum of a winch.
(a) How much work is required to wind the chain onto the drum?
(b) How much work is required to wind the first 30 feet of chain onto the drum?
ANS:
(a) 25,000 ft-lb
(b) 12,750 ft-lb
21. A hemispherical tank with radius 8 feet is filled with water to a depth of 6 feet. Find the work required to empty the tank by pumping the water to the top of the tank.
ANS:
900 ft-lbs, where w is the weight of 1 ft3 of water, in lbs.
22. Suppose a hemispherical tank with radius 10 feet is filled with liquid whose density is 62 pounds per cubic foot. Find the work required to pump all the liquid out through the top of the tank.
ANS:
155,000 ft-lbs
23. A tank 5 feet long has cross-sections n the shape of a parabola (where x and y are in feet). Suppose that the tank is filled to a depth of 3 feet with liquid weighing 15 lb/ft3. How much work is required to empty the tank by pumping the liquid over the edge of the tank?
ANS:
ft-lb
24. The tank in the figure below is full of water with a density 62.5 lb/ft3.
How much work is required to empty the tank by pumping water to a point 4 feet above the top of the tank?
ANS:
25. Find the total hydraulic force on a dam in the shape of an equilateral triangle with one vertex pointing down, if the side of the triangle is 100 feet and the water is even with the top.
ANS:
7,812,500 lb
26. A flat plate of negligible thickness is in the shape of a right triangle with base 5 ft and height 10 ft. The plate is submerged in a tank of water. Find the force on the face of the plate under the following conditions: (Use 62.4 lb/ft3 as the density of water.)
(a) The plate is submerged horizontally so that it rests flat on the bottom of the tank at a depth of 14 ft.
(b) The plate is submerged vertically, base edge up and base at a depth of 3 ft.
ANS:
(a) 21,840 lb
(b) 9880 lb
27. A swimming pool 5 m wide, 10 m long, and 3 m deep is filled with seawater of density 1030 kg/m3 to a depth of 2.5 m. Find
(a) the hydrostatic pressure at the bottom of the pool.
(b) the hydrostatic force on the bottom.
(c) the hydrostatic force on one end of the pool.
ANS:
(a) 25.2 kPa
(b) 1.26 N
(c) 1.58 N
28. A tank contains water. The end of the tank is vertical and has the shape below. Find the hydrostatic force against the end of the tank.
ANS:
(metric units assumed)
29. A tank contains water. The end of the tank is vertical and has the shape below. Find the hydrostatic force against the end of the tank.
ANS:
1.23 N
30. A tank contains water. The end of the tank is vertical and has the shape below. Find the hydrostatic force against the end of the tank.
ANS:
N
31. Two people weighing 100 lb and 180 lb are at opposite ends of a seesaw 14 ft long. Where should the point of support be placed so that the seesaw will balance?
ANS:
The point of support should be 9 feet from the 100 lb person.
32. Calculate the center of mass of a lamina with the given density and shape:
ANS:
(a)
b)
(c)
33. Find the centroid of the region bounded by
ANS:
(x, y) =
34. Find the center of mass of the lamina of uniform density bounded by and x-axis.
ANS:
Center of mass (0, )
35. Determine the centroid of the region bounded by the equation in the first quadrant between x = 1 and x = 4.
ANS:
(2.65, 2.41)
36. City B (population 4,000) is 10 miles north of City A (population 6,000), and City C (population 5,000) is 30 miles east of City B. Where is the best place to locate a super market serving the people in these cities?
ANS:
Let City A be at (0, 0) , City B at (0, 10) and City C at (30, 10)
x = , y =
37. Find the centroid of the region bounded by .
ANS:
38. Find the centroid of the region bounded by .
ANS:
Section 6.7: Applications to Economics and Biology
1. Suppose a company has estimated that the marginal cost of manufacturing x items is (measured in dollars per unit) with a fixed start-up cost of c(0) = 10,000. Find the cost of producing the first 500 items.
a. $15,000 e. $25,000
b. $5,000 f. $6000
c. $10,000 g. $20,000
d. $17,500 h. $60,000
ANS: A
2. Suppose a company has estimated that the marginal cost of manufacturing x pairs of a new line of jeans is (measured in dollars per pair) with a fixed start-up cost of c(0) = 2000. Find the cost of producing the first 1000 pair of jeans.
a. $2000 e. $3000
b. $4000 f. $5000
c. $6000 g. $8000
d. $10,000 h. $15,000
ANS: G
3. The demand function for a certain commodity is Find the consumer surplus when the sales level is 30.
a. 90 e. 30
b. 45 f. 60
c. 15 g. 20
d. 70 h. 80
ANS: C
4. If the demand function for a certain commodity is and the consumer surplus is 15, what should be the production level?
a. 90 e. 30
b. 45 f. 60
c. 15 g. 20
d. 70 h. 80
ANS: E
5. If the demand function for a certain commodity is and the consumer surplus is 15, what should be the sale price?
a. 1 e. 3.3
b. 4 f. 6
c. 1.5 g. 2
d. 2.5 h. 3
ANS: H
6. The demand function for a certain commodity is . Find the consumer surplus when the selling price is $18?
a. 135 e. 270
b. 30 f. 40
c. 45 g. 90
d. 360 h. 180
ANS: G
7. A supply function is given by , where x is the number of units produced. Find the producer surplus when the selling price is $15.
a. 100 e. 800
b. 500 f. 400
c. 250 g. 200
d. 1000 h. 2000
ANS: B
8. A rental estate management company manages an apartment complex with 20 units. The manager estimates that all 20 units can be rented if the rent is $150 per unit per month and that for each increase in rent of $10, one apartment will be vacated. Find the consumer surplus when 5 apartments are vacated.
a. 3000 e. 1125
b. 2250 f. 6000
c. 1250 g. 2500
d. 1000 h. 3500
ANS: E
9. The marginal revenue from selling x items is . The revenue from the sale of the first 100 items is $8800. What is the revenue from the sale of the first 200 items?
ANS:
$17, 500
10. The marginal revenue for a company when sales are q units is given by . Find the increase in revenue when the sales level increases from 100 to 200 units.
ANS:
$300
11. The marginal cost for a company is given by where q is the number of units produced. What is the total cost to raise production from 100 to 200 units?
ANS:
$11,716.60
12. The marginal cost for the production of the new Super Widget at Widget International is give by , whereas the marginal revenue is , where in both cases q represents the number of units produced.
(a) Determine the change in profits when sales are increased from 700 to 1700 units.
(b) What is the change in profit when sales increase from 1700 to 2700 units? Discuss your answer.
ANS:
(a) $1640
(b) . The company should change its production level.
13. The demand function for a certain commodity is Find the consumer’s surplus when the sales level is 30. Illustrate by drawing the demand curve and identifying the consumer’s surplus as an area.
ANS:
$45
14. The demand function for producing a certain commodity is given by . Find the consumer surplus when the sale level is 500.
ANS:
20,833.
15. A manufacture has been selling 1000 ceiling fans at $60 each. A market survey indicates that for every $10 that price is reduced, the number of sets sold will increase by 100. Find the demand function and calculate the consumer surplus when the selling price is set at $50.
ANS:
Demand function , Consumer surplus = $60,500
16. The demand function for a certain commodity is . Find the consumer surplus when the selling price is $18.
ANS:
17. The demand function for a certain commodity is , and consumer surplus is 90. What should be the selling price?
ANS:
Solve for P, such that
18. The dye dilution method is used to measure cardiac output with 6 mg of dye. The dye concentrations, in mg/L, are modeled by , where t is measured in seconds. Find the cardiac output.
ANS:
0.16 L/sec
19. The following table shows the relationship between price and demand for milk produced in a large dairy.
q (billions of pounds of milk per year) 45 50 55 60 65 70 75
p (price in dollars per pound) 1.00 0.90 0.80 0.70 0.60 0.50 0.40
Determine the consumer’s surplus when the sales quality is 65 billion pounds of milk in a year. Illustrate your answer by drawing the corresponding demand curve and the identifying the consumer’s surplus as a region.
ANS:
$42.25
Section 6.8: Probability
1. Find k so that the function can serve as the probability density function of a random variable X.
a. e. 3
b. f. 6
c. g. 2
d. 1 h. 0
ANS: F
2. Find k so that the function can serve as the probability density function of a random variable X.
a. e. 3
b. f. 6
c. g. 2
d. 1 h. 0
ANS: G
3. Let X be a continuous random variable with density function
If the median of this distribution is , then c is:
a. e. 3
b. f.
c. 2 g.
d. h. 3
ANS: D
4. Let X be a continuous random variable with density function
What is the mean of X?:
a. e.
b. 1 f.
c. g.
d. h.
ANS: G
5. Find c so that the following can serve as the probability density function of a random variable X:
a. 8 e. 1
b. 4 f. 16
c. g.
d. h. 2
ANS: A
6. Let be the probability density function of a random variable X. Find the mean of the probability density function .
a. e. 3
b. f. 6
c. g. 2
d. 1 h. 0
ANS: C
7. Let be the probability density function of a random variable X. Find the median of the probability density function .
a. e. 3
b. f. 6
c. g. 2
d. 1 h. 0
ANS: C
8. Let be the probability density function of a random variable X. Find the median of the probability density function .
a. e.
b. f.
c. g.
d. h.
ANS: F
9. (a) Show that is the probability density function of a random variable.
(b) What is the mean for this distribution?
(c) Calculate the median of
ANS:
(a)
(b)
(c) About 0.3774
10. (a) Show that where k is fixed and is a probability density function.
(b) What is the mean for this distribution?
(c) Calculate the median of
ANS:
(a)
(b)
(c)
11. (a) Explain why the function defined by the graph below is a probability density function.
(b) Use the graph to find the following probabilities:
(i)
(ii)
(c) Calculate the median for this distribution.
ANS:
(a)
(b) (i) 0.1
(ii) 0.5
(c) 4.5
12. Assume the daily consumption of electric power (in millions of kilowatt-hours) of a certain city
has the probability density
If the city’s power plant has a daily capacity of 12 million kilowatt-hours, what is the probability that the available power supply will be inadequate on any given day?
ANS:
13. Let be the probability density function of a random variable T, where t is the time that a customer spends in line at teller’s window before being served. What is the probability that a customer will wait more than 10 minutes?
ANS:
14. Let be the probability density function of a random variable T, where t is the time that a customer spends in line at teller’s window before being served. What is mean of the probability density function?
ANS:
15. Let be the probability density function of a random variable T, where t is the time that a customer spends in line at teller’s window before being served. What is median of the probability density function?
ANS:
16. Suppose that the mileage (in thousands of miles) which car owners can obtain from a certain kind of tire has the probability density
Find the probability that a tire chosen at random will last
(a) at most 10,000 miles.
(b) between 15,000 and 25,000 miles.
(c) at least 30,000 miles.
ANS:
(a)
(b)
(c)
17. Assume the weights of adult males are normally distributed with a mean weight of 150 pounds and a standard deviation of 20 pounds. Use Simpson’s Rule or the Midpoint Rule to estimate the following:
(a) What is the probability that an adult male chosen at random will weigh between 120 pounds and 180 pounds?
(b) What percentage of the adult male population weighs more than 200 pounds?
ANS:
(a) About 0.87
(b) About 0.006
18. IQ scores are assumed to be normally distributed with a mean and standard deviation . Use either Simpson’s Rule or the Midpoint Rule to approximate the probability that a person selected at random from the general population will have an IQ score
(a) between 70 and 130.
(b) over 130.
ANS:
(a) About 0.9544
(b) About 0.0228
19. Let
(a) Find k so that f can serve as the probability density function of a random variable X.
(b) Find .
(c) Find the mean.
ANS:
(a) k = 9
(b)
(c)
20. The density function for the life of a certain type of battery is modeled by and is measured in months.
(a) What is the probability that a battery will wear out during the first month of use?
(b) What is the probability that a battery is functioning for more than 12 months?
ANS:
(a) About 0.181
(b) About 0.09
21. The density function for the life of a certain type of battery is modeled by and is measured in months.
(a) Find the median life of the batteries.
(b) Find the mean life of the batteries.
(c) Sketch the graph of the density function showing the median and mean.
ANS:
(a) 5 ln 2 3.466
(b) 5 months
(c)
22. The density function for the waiting time at a bank is modeled by and is measured in minutes.
(a) What is the probability that a customer will be served within the first 5 minutes?
(b) What is the probability that a customer has to wait for more than 15 minutes?
ANS:
(a) About 0.39
(b) About 0.22
23. The density function for the waiting time at a bank is modeled by and is measured in minutes.
(a) Find the median waiting time.
(b) Find the mean waiting time.
(c) Sketch the graph of the density function showing the median and mean.
ANS:
(a) 10 ln 2 6.93 minutes
(b) 10 minutes
(c)
Section 7.1: Modeling with Differential Equations
1. Which of the following equations is satisfied by the function ?
a. e.
b. f.
c. g.
d. h.
ANS: C
2. Which of the following equations is satisfied by the function ?
a. e.
b. f.
c. g.
d. h.
ANS: G
3. Which of the following equations is satisfied by the function ?
a. e.
b. f.
c. g.
d. h.
ANS: B
4. Which of the following is a solution of the differential equation
a. e.
b. f.
c. g.
d. h.
ANS: A
5. Which of the following is a solution of the differential equation
a. e.
b. f.
c. g.
d. h.
ANS: H
6. Which of the following is a solution of the differential equation
a. e.
b. f.
c. g.
d. h.
ANS: G
7. Which of the following is a solution of the differential equation
a. e.
b. f.
c. g.
d. h.
ANS: C
8. Which of the following is a solution of the differential equation
a. e.
b. f.
c. g.
d. h.
ANS: D
9. Which of the following is a solution of the differential equation
a. e.
b. f.
c. g.
d. h.
ANS: E
10. Which of the following is a solution of the differential equation
a. e.
b. f.
c. g.
d. h.
ANS: F
11. Which of the following is a solution of the differential equation
a. e.
b. f.
c. g.
d. h.
ANS: B
12. Which of the following is a solution of the differential equation
a. e.
b. f.
c. g.
d. h. None of these
ANS: D
13. Which of the following is a solution of the differential equation ?
a. e.
b. f.
c. g.
d. h. None of these
ANS: E
14. Which of the following is a solution of the differential equation that satisfies the initial conditions , , ?
a. e.
b. f.
c. g.
d. h. None of these
ANS: B
15. (a) Show that every member of the family of functions is a solution of the differential equation .
(b) Find a solution of the differential equation that satisfies the initial condition .
(c) Find a solution of the differential equation that satisfies the initial condition .
ANS:
(a)
(b)
(c)
16. (a) What can you conclude about the functions which satisfy just by looking at the differential equation?
(b) Verify that are solutions of the equation in part (a).
(c) Is there a solution of the equation in part (a) that is not a member of the family of functions in part (b)? Justify your answer.
(d) Find a solution to the equation in part (a) with the additional condition that .
ANS:
(a) is a constant solution and if , . Therefore solutions of the differential equation are increasing functions.
(b)
(c) is not included in part (b).
(d)
17. (a) What can you conclude about the graph of the solution of the equation just by looking at the differential equation?
(b) Use implicit differentiation to verify all members of the family are solutions of the equation in part (a).
(c) Find a solution of the equation in part (a) with the additional condition that .
ANS:
(a) Let be a graph of a solution of the differential equation. Then , so the slope of the tangent of is perpendicular to the line passing through and .
(b)
(c)
18. A function y (x) satisfies the differential equation .
(a) What are the constant solutions of the equation?
(b) For what values of is increasing?
(c) The equation shows that is independent of . Using this observation, what can you say about the relationship of the graphs of the non-constant solutions of the equation?
ANS:
(a) and .
(b) For or , is increasing.
(c) The slopes corresponding to two different points with the same y-coordinates must be equal. This means that if we know one solution to the equation, then we can obtain infinitely many others just by shifting the graph of the known solution to the right or left.
19. The study of free fall provides one context to consider differential equations. In the simplest case, in the absence of air or other resistance, physicists assume that the rate of change of velocity of a body is constant.
(a) Write an equation for .
(b) As the time increases without bound, what happens to the velocity ?
ANS:
(a) where is a constant.
(b) Since , where is a constant, therefore as increases without bound, will increase without bound.
20. The study of free fall provides one context to consider differential equations. In the simplest case, in the absence of air or other resistance, physicists assume that the rate of change of velocity of a body is constant. But it is more realistic to consider the presence of air resistance. Assume that is the constant of acceleration due to earth’s gravity. Suppose that air resistance is proportional to the velocity of the falling body.
(a) Explain why the differential equation , where is a positive constant, would be a reasonable model for velocity under these conditions.
(b) When does increase most rapidly? Justify your answer.
(c) Consider the equation in part (a). What would happen to the rate of change of velocity, , as increases? Justify your conclusion.
(d) Make a sketch of a possible solution for this differential equation.
ANS:
(a) shows that the ratio at which the velocity increases, decreases at a rate proportional to the velocity.
(b) for . Therefore increases most rapidly at .
(c) As increases, will approach 0. That means the force of resistance is just equal to the gravity. Therefore, the velocity will approach a limiting value.
(d)
21. In paleontology, the phenomenon of radioactive decay is commonly used to date fossil remains. This method uses the fact that the rate of decay of an element will be proportional to the amount of the radioactive element that exists at time .
(a) Determine a differential equation that must satisfy.
(b) What can you say about as ?
(c) Find a solution to your differential equation.
(d) How long will it take for half of the original amount of the radioactive element to decay?
ANS:
(a) , where is a positive constant.
(b) Since , will approach as .
(c)
(d)
22. In biology, it is often assumed that the number of people that will contract a certain disease is directly proportional to the product of the number of people who are currently infected with the disease at a particular time and the number of people who, at the same time, are not yet infected but who are susceptible to it. Assume that the total population, , in the study is a constant.
(a) What does mean?
(b) Determine a differential equation that must satisfy.
(c) What happens to as ?
ANS:
(a) is the infection rate.
(b)
(c) Since , as , .
23. A ball is thrown directly upward from the ground with an initial speed ft/s.
(a) Determine the differential equation that the position function satisfies.
(b) What are the initial conditions?
(c) How high does the ball rise?
ANS:
(a)
(b) ,
(c) feet
24. The brakes of a car traveling decelerate the car at the rate of 20 ft/s2.
(a) Determine the differential equation that the position function satisfies.
(b) What are the initial conditions?
(c) If the car is from a barrier when the brakes are applied, will it hit the barrier?
ANS:
(a)
(b) ,
(c) Yes, because the care needs in order to stop.
25. A tank is filled with 100 L (liter) of water. Brine containing 0.4 kg of salt per liter is poured into the tank at 5 L per minute, and the well-stirred mixture leaves the tank at the same rate.
(a) Write the differential equation that the amount of salt, in the tank must satisfy.
(b) Find the amount of salt in the tank at any time, .
(c) How much salt is present after 20 min?
(d) What is the concentration of solution in the tank after a long time?
ANS:
(a) ,
(b)
(c)
(d) per liter
26. Suppose a hail ball melts at a rate proportional to its surface area. If from to minutes its radius decreases from cm to cm, how long does it take to melt completely?
ANS:
where represents the volume of the hail ball and represents the surface area of the ball.
We know that
, and ,
The hail ball melted completely, minutes.
Section 7.2: Direction Fields and Euler’s Method
1. A direction field is given below. Which of the following represents its differential equation?
a. e.
b. f.
c. g.
d. h.
ANS: G
2. A direction field is given below. Which of the following represents its differential equation?
a. e.
b. f.
c. g.
d. h.
ANS: F
3. A direction field is given below. Which of the following represents its differential equation?
a. e.
b. f.
c. g.
d. h.
ANS: E
4. A direction field is given below. Which of the following represents its differential equation?
a. e.
b. f.
c. g.
d. h.
ANS: D
5. A direction field is given below. Which of the following represents its differential equation?
a. e.
b. f.
c. g.
d. h.
ANS: B
6. A direction field is given below. Which of the following represents its differential equation?
a. e.
b. f.
c. g.
d. h.
ANS: A
7. Which of the given differential equations matches the given direction field?
a. e.
b. f.
c. g.
d. h.
ANS: B
8. Which of the given differential equations matches the given direction field?
a. e.
b. f.
c. g.
d. h.
ANS: G
9. Which of the given differential equations matches the given direction field?
a. e.
b. f.
c. g.
d. h.
ANS: C
10. Which of the given differential equations matches the given direction field?
a. e.
b. f.
c. g.
d. h.
ANS: D
11. Suppose , . Use Euler’s method with step size to approximate .
a. 5 e. 1
b. 2 f. 4
c. 3 g. 9
d. 17 h. 16
ANS: G
12. Suppose , . Use Euler’s method with step size to approximate .
a. 1 e. 4
b. 8 f. 14
c. 16 g. 12
d. 9 h. 6
ANS: D
13. Suppose , . Use Euler’s method with step size to approximate .
a. 10 e. 1
b. 7.5 f. 2.5
c. 9.5 g. 4
d. 5 h. 6
ANS: H
14. Suppose , . Use Euler’s method with step size to approximate .
a. 3.6 e. 1
b. 3.2 f. 4.312
c. 3.4 g. 4
d. 5.14 h. 6.1
ANS: F
15. Suppose , . Use Euler’s method with step size to approximate .
a. 28 e. 30
b. 8 f. 18
c. 16 g. 15
d. 3 h. 6
ANS: C
16. Suppose , . Use Euler’s method with step size to approximate .
a. 24 e. 25
b. 8 f. 14
c. 16 g. 12
d. 4 h. 6
ANS: A
17. Suppose , . Use Euler’s method with step size to approximate .
a. 3.5 e. 2.5
b. 5.5 f. 6
c. g.
d. h.
ANS: D
18. A direction field for a differential equation is given below:
(a) Sketch the graphs of the solutions that have initial condition and initial condition .
(b) Determine whether the differential equation is autonomous. Explain your answer.
ANS:
(a)
(b) It is autonomous since is independent of .
19. A direction field for a differential equation is given below:
(a) Sketch the graphs of the solutions that have initial condition and initial condition .
(b) Determine whether the differential equation is autonomous. Explain your answer.
ANS:
(a)
(b) It is autonomous since is constant and independent of .
20. A direction field for a differential equation is given below:
(a) Sketch the graphs of the solutions that have initial condition and initial condition .
(b) Determine whether the differential equation is autonomous. Explain your answer.
ANS:
(a)
(b) It is not autonomous since is not independent of .
21. A direction field for a differential equation is given below:
(a) Sketch the graphs of the solutions that have initial condition and initial condition .
(b) Determine whether the differential equation is autonomous. Explain your answer.
ANS:
(a)
(b) It is not autonomous since is dependent on both and .
22. A direction field for a differential equation is given below:
(a) Sketch the graphs of the solutions that have initial condition and initial condition .
(b) Determine whether the differential equation is autonomous. Explain your answer.
ANS:
(a)
(b) It is autonomous since is independent of .
23. A population is modeled by the differential equation where is the population at time .
(a) What are the equilibrium solutions?
(b) For what values of is the population increasing?
(c) For what values of is the population decreasing?
(d) Use the information from above to sketch the direction field for the given differential equation.
ANS:
(a) ,
(b)
(c)
(d)
24. Consider the differential equation .
(a) What are the equilibrium solutions?
(b) For what values of is increasing?
(c) For what values of is decreasing?
(d) Use the information from above to sketch the direction field for the given differential equation.
ANS:
(a) , ,
(b) or
(c) or
(d)
25. Consider the differential equation .
(a) What are the equilibrium solutions?
(b) Sketch the direction field for the given differential equation.
ANS:
(a) None
(b)
26. Consider the differential equation .
(a) What are the equilibrium solutions?
(b) Sketch the direction field for the given differential equation.
ANS:
(a) None
(b)
27. Consider the differential equation .
(a) What are the equilibrium solutions?
(b) What are the points in the -plane at which the slope of the solution curve is ?
(c) What are the points in the -plane at which the slope of the solution curve is ?
(d) What are the points in the -plane at which the slope of the solution curve is –1?
(e) Use the information from above to sketch the direction field for the given differential equation.
ANS:
(a) None
(b)
(c)
(d)
(e)
28. Make a rough sketch of a directional field for the autonomous differential equation where is the function graphed below.
ANS:
29. Make a rough sketch of a directional field for the autonomous differential equation where is the function graphed below.
ANS:
30. (a) Determine the solution of the differential equation where .
(b) Use the solution from part (a) to calculate .
(c) Use Euler’s Method with the given step sizes to estimate the value of for the equation given in part (a).
(i)
(ii)
(iii)
(d) Sketch from part (b) and each of the Euler approximations from part (c) on the same set of axes.
ANS:
(a)
(b)
(c) (i) and
(ii) , ,
(iii) , , , ,
(d)
31. (a) Determine the solution of the differential equation where .
(b) Use the solution from part (a) to calculate .
(c) Use Euler’s Method with the given step sizes to estimate the value of for the equation given in part (a).
(i)
(ii)
(iii)
(d) Sketch from part (b) and each of the Euler approximations from part (c) on the same coordinate plane.
ANS:
(a)
(b)
(c) (i) and
(ii) and ,
(iii) and , , ,
(d)
32. A direction field for a differential equation is given below. Use a straightedge to draw the graphs of the Euler approximations to the solution curve over the interval that passes through . Use as step sizes , , and .
ANS:
33. A direction field for a differential equation is given below. Use a straightedge to draw the graphs of the Euler approximations to the solution curve over the interval that passes through . Use as step sizes , , and .
ANS:
34. A direction field for a differential equation is given below. Use a straightedge to draw the graphs of the Euler approximations to the solution curve over the interval that passes through . Use as step sizes , , and .
ANS:
35. Consider the differential equation .
(a) Sketch the direction field. Indicate where the slopes are , 0, or 1. Draw some other slopes as well.
(b) If the point is on the graph of a solution, use Euler’s Method with step size to estimate the value of the solution at .
ANS:
(a)
(b)
36. Consider the differential equation .
(a) Sketch the direction field. Indicate where the slopes are , 0, or 1. Draw some other slopes as well.
(b) If the point is on the graph of a solution, use Euler’s Method with step size to estimate the value of the solution at .
ANS:
(a)
(b)
Section 7.3: Separable Equations
1. Find the solution of the initial-value problem , .
a. e.
b. f.
c. g.
d. h.
ANS: C
2. Solve the differential equation subject to the initial condition . From your solution, and the value of the limit .
a. 5000 e. 200
b. 2500 f. 20000
c. 1000 g. 100
d. 2000 h. 500
ANS: C
3. Solve the differential equation , . From your solution, and the value of .
a. e.
b. 1 f. 3
c. g.
d. h.
ANS: E
4. Solve the differential equation , . From your solution, and the value of .
a. e.
b. 2 f. 3
c. 5 g.
d. 0 h.
ANS: F
5. Find the solution of the initial-value problem , .
a. e.
b. f.
c. g.
d. h.
ANS: G
6. Find the solution of the initial-value problem , .
a. e.
b. f.
c. g.
d. h.
ANS: D
7. Find the solution of the initial-value problem , .
a. e.
b. f.
c. g.
d. h.
ANS: D
8. Solve the initial-value problem , . Then use your solution to evaluate .
a. e. 0
b. 1 f. 3
c. g.
d. h.
ANS: C
9. Find the solution of the initial-value problem , .
a. e.
b. f.
c. g.
d. h.
ANS: F
10. Solve the initial-value problem , . Then use your solution to evaluate .
a. e.
b. f.
c. g. 12
d. 1 h. 27
ANS: A
11. Find the solution of the initial-value problem , .
a. e.
b. f.
c. g.
d. h.
ANS: G
12. Solve the initial-value problem , . Then use your solution to evaluate .
a. e.
b. f.
c. g.
d. h. 9
ANS: F
13. Find the solution of the initial-value problem , .
a. e.
b. f.
c. g.
d. h.
ANS: H
14. Solve the initial-value problem , .
a. e.
b. f.
c. g.
d. h.
ANS: E
15. Consider the differential equation .
(a) Find the general solution to the differential equation.
(b) Find the solution with the initial-value .
(c) Find the solution with the initial-value .
ANS:
(a)
(b)
(c)
16. Consider the differential equation .
(a) Find the general solution to the differential equation.
(b) Find the solution that satisfies the initial condition .
ANS:
(a)
(b)
17. Find the equation of a curve that passes through the point and whose slope at a point is .
ANS:
18. Consider the differential equation .
(a) Find the general solution to the differential equation.
(b) Find the solution that satisfies the initial condition .
ANS:
(a)
(b)
19. Find the solution to the differential equation that satisfies the initial condition .
ANS:
20. Find the solution to the differential equation that satisfies the initial condition .
ANS:
21. The graph of a direction field for the differential equation is given below:
(a) Sketch a solution curve that satisfies the given condition, but without solving the differential equation:
(i)
(ii)
(iii)
(b) Solve the differential equation for each of the conditions in part (a). Compare your answers to the curves you produced in part (a).
(c) What is the relationship between the curves (i) and (ii) in part (a)? Explain why this occurs.
ANS:
(a)
(i) y(0) = 1 (ii) y(2) = 1 (ii) y(0) = 0
(b)
(c) The solutions to (i) and (ii) are the same.
22. Find the orthogonal trajectories of the family of curves . Then draw several members
of each family on the same coordinate plane.
ANS:
23. Find the orthogonal trajectories of the family of curves . Then draw several members of each family on the same coordinate plane.
ANS:
24. Find the orthogonal trajectories of the family of curves . Then draw several members of each family on the same coordinate plane.
ANS:
25. In the presence of air resistance for an object in free fall, the velocity is the solution of the
differential equation where is the constant of acceleration due to earth’s gravity and is a positive constant.
(a) Find the solution for the equation that satisfies the initial value that .
(b) As time, , increases without bound, what is the limiting velocity. (Note: the limiting velocity during free fall is called the terminal velocity.)
(c) Sketch your solution from part (a).
ANS:
(a)
(b)
(c)
26. In a model of epidemics, let , in thousands, be the number of infected individuals in the
population at time , in days. If we assume that the infection spreads to all those who are
susceptible, one possible solution for is given by where is a positive
constant which measures the rate of infection and , in thousands, is the total population in
this situation.
(a) Determine the solution of this differential equation if .
(b) Discuss what means.
(c) As the time increases without bound, what happens to ? (That is, what does mean?)
(d) Sketch the solution of the differential equation in part (a).
ANS:
(a)
(b) At initial time there were 1000 infected individuals.
(c) , which is the total population in this situation.
(d)
27. A model of the seasonal changes in daylight hours is given by , where
and are constants.
(a) Determine the general solution for this differential equation.
(b) Discuss the practical meaning of the solution of this equation.
ANS:
(a)
(b) The solutions are periodic functions which model seasonal change in daylight hours.
28. A tank is filled with 200 gallons of brine in which is dissolved 5 pounds of salt. Brine containing 0.1 pound of salt per gallon enters the tank at a rate of 2 gallons per minute, and the well-stirred mixture is drawn from the talk at the same rate.
(a) Find the amount of salt in the tank at time .
(b) How much salt is present in the tank after 20 minutes?
(c) How much salt is present after a long time? What is the concentration then?
ANS:
(a)
(b) 8.33 lb
(c) 40 lb; 20%
29. A tank contains 10 gallons of water and 4 pounds of a chemical, , per gallon. To decrease the concentration of , pure water is added to the container at a rate of 2 gallons per minute, and the well stirred mixture is drawn from the tank at the same rate.
(a) Find the amount of chemical in the tank at time .
(b) How much chemical is present after 10 minutes?
(c) How long does it take for the concentration of chemical to be reduced to 0.1 pound per gallon?
ANS:
(a)
(b) 0.073 lb
(c) 6.93 min
30. A tank contains 500 liters of brine with 10 kg of dissolved salt. Pure water enters the tank at a rate of 10 liters per minute. The solution is kept thoroughly mixed and drains from the tank at the same rate. How much salt is in the tank:
(a) after minutes?
(b) after 10 minutes?
ANS:
(a)
(b) 8.19 kg
31. A tank contains 500 liters of brine with 10 kg of dissolved salt. Pure water enters the tank at a rate of 10 liters per minute. The well-mixed solution drains from the tank 8 liters per minute. Determine the differential equation that the amount of salt in the tank at time must satisfy.
ANS:
with initial condition
32. A tank contains 1000 liters of pure water. Brine that contains 0.05 kg of salt per liter of water enters the tank at a rate of 8 liters per minute. Brine that contains 0.04 kg of salt per liter of water enters the tank at a rate of 5 liters per minute. The well-mixed solution drains from the tank at a rate of 13 liters per minute. How much salt is in the tank after:
(a) minutes?
(b) half an hour?
(c) a very long time?
ANS:
(a)
(b) 3.97 kg
(c) kg
33. The discharge value on a 1000 liter tank that is filled with water is opened at time t = 0 and the water flows out at a rate of 10 liters per second. At the same time a 1% chlorine mixture enters the tank at a rate of 6 liters per second. Assuming that the solution is well-mixed throughout the tank, what is the concentration of chlorine when the tank is half full?
ANS:
34. Newton’s Law of Cooling states that the rate at which a body changes temperature is proportional to the difference between its temperature and the temperature of the surrounding medium. Suppose that a body has an initial temperature of 250 F and that after one hour the temperature is 200 F. Assuming that the surrounding air is kept at a constant temperature of 72 F, determine the temperature of the body at time .
ANS:
35. According to Newton’s Law of Cooling, the temperature of a warm object decreases at a rate proportional to the difference between and the temperature of its surroundings.
(a) Write down this law as a differential equation.
(b) Assume the room temperature is 70 F. If it takes 2 minutes for a cup of hot coffee to cool down to 180 F, and how long it takes to cool a cup of coffee from 200 F to 100 F.
ANS:
(a)
(b) , 17.56 minutes
36. According to Newton’s Law of Heating, the temperature of a cold object increases at a rate proportional to the difference between and the temperature of its surroundings.
(a) Write down this law as a differential equation.
(b) Assume the room temperature is 70 F. If it takes 10 minutes for a can of soda to warm up from 30 F to 35 F, and how long it takes to warm up a can of soda from 30 F to 40 F.
ANS:
(a)
(b) 21.5 minutes
Section 7.4: Exponential Growth and Decay
1. The radioactive isotope Bismuth-210 has a half-life of 5 days. Suppose we have an initial amount of 100 mg. The amount of Bismuth-210 remaining after days is
a. e.
b. f.
c. g.
d. h.
ANS: D
2. The radioactive isotope Bismuth-210 has a half-life of 5 days. How many days does it take for 87.5% of a given amount to decay?
a. 15 days e. 11 days
b. 8 days f. 9 days
c. 10 days g. 12 days
d. 13 days h. 14 days
ANS: A
3. A bacteria culture starts with 200 bacteria and triples in size every half hour. The population of the bacteria after hours is:
a. e.
b. f.
c. g.
d. h.
ANS: B
4. A bacteria culture starts with 200 bacteria and triples in size every half hour. After 2 hours, how many bacteria are there?
a. 17,800 e. 19,300
b. 16,200 f. 14,800
c. 23,500 g. 15,700
d. 24,000 h. 21,000
ANS: B
5. A bacteria culture starts with 200 bacteria and in 1 hour contains 400 bacteria. How many hours does it take to reach 2000 bacteria?
a. e.
b. f.
c. g.
d. h.
ANS: H
6. When a child was born, her grandparents deposited $1000 in a saving account at 5% interest compounded continuously. The amount of money after t years is:
a. e.
b. f.
c. g.
d. h.
ANS: F
7. When a child was born, her grandparents placed $1000 in a savings account at 10% interest compounded continuously, to be withdrawn at age 20 to help pay for college. How much money is in the account at the time of withdrawal?
a. e.
b. f.
c. g.
d. h.
ANS: G
8. Radium has a half-life of 1600 years. How many years does it take for 90% of a given amount of radium to decay?
a. e.
b. f.
c. g.
d. h.
ANS: C
9. Carbon 14, with a half-life of 5700 years, is used to estimate the age of organic materials. What fraction of the original amount of carbon 14 would an object have if it were 2000 years old?
a. e.
b. f.
c. g.
d. h.
ANS: C
10. The half-life of Carbon 14 is 5700 years. A wooden table is measured with 80% of Carbon 14 compared with newly cut tree. Find the age of the table.
a. 2, 933 years e. 13,235 years
b. 1,000 years f. 4,200 years
c. 500 years g. 1,835 years
d. 2,000 years h. 3,000 years
ANS: G
11. An object cools at a rate (measured in ) equal to times the difference between its temperature and that of the surrounding air. Suppose the object takes 10 minutes to cool from 60 C to 40 C in a room kept at 20 C. Find the value of .
a. e.
b. f.
c. g.
d. h.
ANS: G
12. A bacteria population grows at a rate proportional to its size. The initial count was 400 and 1600 after 1 hour. In how many minutes does the population double?
a. 20 e. 40
b. 25 f. 45
c. 30 g. 50
d. 35 h. 55
ANS: C
13. An object cools at a rate (in ) equal to of the difference between its temperature and that of the surrounding air. If a room is kept at 20 C and the temperature of the object is 28 C, what is the temperature of the object 5 minutes later?
a. 22 e.
b. 24 f.
c. g.
d. h.
ANS: D
14. A thermometer is taken outside from a room where the temperature is 72 F. Outdoors, the temperature is 48 F. After one minute, the thermometer reads 55 F. After how many minutes does the thermometer read 50 F?
a. 2.107 e. 2.017
b. 1.107 f. 1.017
c. 3.100 g. 3.010
d. 1.503 h. 1.013
ANS: E
15. 1 cent is invested at 6% annual interest, compounded continuously. Let A(t) be the amount of the investment at time t, measured in years. Find A(200).
a. $1,627.55 e. $162.76
b. $16.28 f. $162,755
c. $160 g. $150
d. $140 h. $130
ANS: A
16. The growth of a population is modeled by the differential equation , and the initial population is Find
a. 37 e. 30
b. 90 f. 80
c. 44,053 g. 90,000
d. 81,350 h. 37,648
ANS: D
17. In an experiment, a tissue culture has been subjected to ionizing radiation. It was found that the number A of undamaged cells depends on the exposure time, in hours, according to the formula
If 5000 cells were present initially and 3000 survived a 2-hour exposure, and the elapsed time of exposure after which only half the original cells survive.
ANS:
2.71 hours
18. A lettuce leaf collected from the salad bar at the college cafeteria contains as much carbon-14 as a freshly cut lettuce leaf. How old is it? (Use 5700 years for the half-life of C.)
ANS:
About 83 years old.
19. Assume that the rate of growth of a population of fruit flies is proportional to the size of the population at each instant of time. If 100 fruit flies are present initially and 200 are present after 5 days, how many will be present after 10 days?
ANS:
400
20. Assume the half-life of carbon 14 is 5700 years. A wooden statue is measured with 70% of the carbon-14. How old is the statue?
ANS:
About 2933 years old.
21. It takes money 20 years to triple at a certain rate of interest. How long does it take for money to double at this rate?
ANS:
12.62 years
22. In 1970, the Brown County groundhog population was 100. By 1980, there were 900 groundhogs in Brown County. If the rate of population growth of these animals is proportional to the population size, how many groundhogs might one expect to see in 1995?
ANS:
24,300
23. In a certain medical treatment, a tracer dye is injected into a human organ to measure its function rate and the rate of change of the amount of dye is proportional to the amount present at any time. If a physician injects 0.5 g of dye and 30 minutes later 0.1 g remains, how much dye will be present in hours?
ANS:
0.004 g
24. In an idealized experiment, the following results were obtained for a population of bacteria during a 7 hour period. The initial population is 1000 bacteria.
(a) Identify the period where there is no change in the number of bacteria. (This is called the period of adaptation.)
(b) Identify the period of growth.
(c) Assume that the growth rate of bacteria is proportional to the population. Find an exponential model for the data during the period of growth.
(d) Add an additional line to the table using your population model to generate the entries for the given time values. Compare these entries with the given data and explain any discrepancy.
ANS:
(a)
(b)
(c) (Answer may vary.)
(d)
25. The following data approximate the results obtained by subjecting Hela-S cells to 250 kvp x-rays:
Assume that these data fit an exponential model.
(a) Find the appropriate exponential model.
(b) Add another line to the table using your population model for the given doses of radiation.
(c) Compare the model entries to the given data and explain any discrepancy.
ANS:
(a)
(b)
(c) Answers may vary. The model does not fit well at , but fits better for larger doses.
26. $2000 is invested at 3% annual interest. Find the value of A(t) at the end of t years if:
(a) the interest compounds annually.
(b) the interest compounds continuously.
ANS:
(a)
(b)
27. $2000 is invested at 3% annual interest. Find the value at the end of 10 years if:
(a) the interest compounds annually.
(b) the interest compounds continuously.
ANS:
(a) About $2687.83
(b) About $2699.72
28. $2000 is invested at 5% annual interest. Find the value of A(t) at the end of t years if:
(a) the interest compounds monthly.
(b) the interest compounds continuously.
ANS:
(a)
(b)
29. $2000 is invested at 5% annual interest. Find the value at the end of 18 years if:
(a) the interest compounds monthly.
(b) the interest compounds continuously.
ANS:
(a) About $4910.02
(b) About $4919.21
Section 7.5: Logistic Equation
1. Suppose a population growth is modeled by the logistic equation . What is the carrying capacity?
a. 90 e. 100
b. 10 f. 60
c. 50 g. 20
d. 1000 h. 10,000
ANS: E
2. Suppose a population growth is modeled by the logistic equation with P(0) = 10. Find the formula for the population after t years.
a. e.
b. f.
c. g.
d. h.
ANS: A
3. Suppose a population growth is modeled by the logistic equation with P(0) = 10. Find the population after 500 years.
a. 50 e. 80
b. 94 f. 100
c. 70 g. 35
d. 500 h. 30
ANS: B
4. Suppose a population growth is modeled by the logistic equation . What is the relative growth rate?
a. 0.0001 e. 0.0002
b. -0.01 f. -0.02
c. 0.001 g. 0.002
d. 0.01 h. 0.02
ANS: D
5. Suppose a population growth is modeled by the logistic equation with P(0) = 10. Find the formula for the population after t years.
a. e.
b. f.
c. g.
d. h.
ANS: G
6. Suppose a population growth is modeled by the logistic equation with P(0) = 10. Find the population after 50 years.
a. 50 e. 80
b. 500 f. 1000
c. 600 g. 350
d. 700 h. 300
ANS: C
7. Suppose a population growth is modeled by the logistic differential equation with the carrying capacity 2000 and the relative growth rate k = 0.06 per year. If the initial population is P(0) = 500, and P(10).
a. 309 e. 308
b. 756 f. 755
c. 310 g. 307
d. 757 h. 800
ANS: B
8. Suppose a population growth is modeled by the logistic equation . Solve this differential equation with the initial condition P(0) = 20.
ANS:
9. Suppose that a population of bacteria grows according to the logistic equation , where P is the population measured in thousands and t is time measured in days.
(a) What is the carrying capacity? What is the value of k?
(b) A direction field for this equation is given below. Where are the slopes close to 0? Where are the slope values the largest? Where are the solutions increasing? Where are the solutions decreasing?
(c) Use the direction field to sketch solutions for initial populations of 10, 30, 50, and 70. What do these solutions have in common? How do they differ? Which solutions have inflection points? At what population levels do they occur?
(d) What are the equilibrium solutions? How are the other solutions related to these solutions?
ANS:
(a) 50 thousand;
(b) Near P = 0 or P = 50 thousand, the slopes are close to 0. At P = 25 thousand, the slope value is the largest. The solutions are increasing for . The solutions are decreasing for .
(c) The solution curves that start below P = 50 are increasing and those that start above P = 50 are decreasing. The solution curves that start below P = 25 have an inflection point when P = 25.
(d) and . The solutions move away from and move toward .
10. A rumor tends to spread according to the logistic differential equation , where y is the number of people in the community who have heard the rumor and t is the time in days.
(a) Describe the population for this sociological study.
(b) Assume that there were 10 people who knew the rumor at initial time t = 0. Find the solution for the differential equation.
(c) How many days will it take for half of the population to hear the rumor?
ANS:
(a) where is the population who heard the rumor when .
(b)
(c) 19 days
11. In a model of epidemics, the number of infected individuals in a population at a time is a solution of the logistic differential equation , where y is the number of infected individuals in the community and t is the time in days.
(a) Describe the population for this situation.
(b) Assume that 10 people were infected at the initial time t = 0. Find the solution for the differential equation.
(c) How many days will it take for half of the population to be infected?
ANS:
(a) where is the number infected when .
(b)
(c) 9.5 days
12. The following table contains population data for a Minnesota county for the decades from 1900 to 1980:
(a) Produce a scatter plot for the data.
(b) Find an exponential model using the data from 1900 through 1950.
(c) Find a logistic model using the data from 1900 through 1950. (Assume the carrying capacity is 440,000.)
(d) Use your models to estimate the population for 1960, 1970, and 1980. Enter your data in the table provided above.
ANS:
(a)
(b) (Answers may vary.)
(c) , where is the year since 1900. (Answers may vary.)
(d)
Note: Answers for (b) and (c) may vary. None of the models is a good fit in this situation. The aim for the problem is to practice the procedure of using data to fit the exponential and logistic models.
13. An outbreak of a previously unknown influenza occurred on the campus of the University of Northern South Dakota at Roscoe during the first semester. Due to the contagious nature of the disease, the campus was quarantined and the disease was allowed to run its course. The table below shows the total number P of infected students for the first four weeks of the outbreak on this campus of 2,500 students.
(a) Find a logistic model for the data. Complete the table with predicted values using this model.
(b) Find an exponential model for these data. Complete the table with predicted values using this model.
(c) Compare your findings in parts (a) and (b) above. For what values would you consider both models to be a good fit for the data? Which model provides the best fit for the data? Justify your choice.
ANS:
(a) (Answers may vary.)
(b) (Answers may vary.)
(c)
Both models fit well for . For , the logistic model fits better.
14. Suppose that a certain population grows according to an exponential model.
(a) Write the differential equation for this situation with a relative growth rate of k = 0.01. Produce a solution for the initial condition t = 0 (in hours) and population P = 1 (in thousands).
(b) Find the population when t = 10 hours, t = 100 hours, and t = 1000 hours.
(c) After how many hours does the population reach 2 thousand? 30 thousand? 55 thousand?
(d) As the time t increases without bound, what happens to the population?
(e) Sketch the graph of the solution of the differential equation.
ANS:
(a)
(b) 1.105 thousand, 2.718 thousand, 22,026.466 thousand
(c) The population will reach 2 thousand in about 69.3 hours. The population will reach 30 thousand in about 340.12 hours. The population will reach 55 thousand in about 400.74 hours.
(d)
(e)
15. Suppose that a population grows according to a logistic model.
(a) Write the differential equation for this situation with k = 0.01 and carrying capacity of 60 thousand.
(b) Solve the differential equation in part (a) with the initial condition t = 0 (hours) and population P = 1 thousand.
(c) Find the population for t = 10 hours, t = 100 hours, and t = 1000 hours.
(d) After how many hours does the population reach 2 thousand? 30 thousand? 55 thousand?
(e) As the time t increases without bound, what happens to the population?
(f) Sketch the graph of the solution of the differential equation.
ANS:
(a)
(b)
(c)
(d) The population will reach 2 thousand in about 71.03 hours. The population will reach 30 thousand in about 407.75 hours. The population will reach 55 thousand in about 647.55 hours.
(e) As thousand
(f)
16. Suppose that a population, P, grows at a rate given by the equation , where P is the population (in thousands) at time t (in hours), and b and k are positive constants.
(a) Find the solution to the differential equation when b = 0.04, k = 0.01 and P (0) = 1.
(b) Find P (10), P (100), and P (1000).
(c) After how many hours does the population reach 2 thousand? 30 thousand? 54 thousand?
(d) As time t increases without bound, what happens to the population?
(e) Sketch the graph of the solution of the differential equation.
ANS:
(a)
(b)
(c) The population will reach 2 thousand in about 19.3 hours. The population will reach 30 thousand in about 190 hours. The population will reach 54 thousand in about 590.3 hours.
(d) As (thousand)
(e)
17. Assume that a population grows at a rate summarized by the equation , where b and k are positive constants (b > 1), and P is the population at time t. Show that is the general solution for the differential equation (where is the initial population). [Note: This is known as the monomolecular growth curve.]
ANS:
Either solve the equation or substitute into the equation to verify. Note: When then P = 0 for this model.
18. (a) Solve the differential equation , with b = 2 and k = 0.1, and = 1.
(b) Sketch a graph of the solution you produced for part (a) and discuss the major characteristics of this monomolecular growth curve.
ANS:
(a)
(b)
P (t) is a decreasing function and when , P = 0.
Section 7.6: Predator-Prey Systems
1. Suppose that we model populations of aphids and ladybugs with the system of differential equations:
Find the equilibrium solution.
a. e.
b. f.
c. g.
d. h.
ANS: A
2. Suppose that we model populations of aphids and ladybugs with the system of differential equations:
Find the expression for .
a. e.
b. f.
c. g.
d. h.
ANS: D
3. Suppose that we model populations of predators and preys (in millions) with the system of differential equations:
Find the equilibrium solution.
a. e.
b. f.
c. g.
d. h.
ANS: E
4. Suppose that we model populations (in millions) of predators and preys with the system of differential equations:
Find the expression for .
a. e.
b. f.
c. g.
d. h.
ANS: F
5. A predator-prey system is modeled by the system of differential equations , , where a, b, c, and d are positive constants.
(a) Which variable, x or y, represents the predator? Defend your choice.
(b) Show that the given system of differential equations has the two equilibrium solutions and .
(c) Explain the significance of each of the equilibrium solutions.
ANS:
(a) represents the predator. In the absence of prey, the predators will die out.
(b) Solve to get these solutions.
(c) and implies that there is neither predator nor prey. The population of both predator and prey are not changing if
6. Consider the predator-prey system , where x and y are in millions of creatures and t represents time in years.
(a) Find equilibrium solutions for this system.
(b) Explain why it is reasonable to approximate this predator-prey system as , if the initial conditions are x(0) = 0.001 and y(0) = 0.002.
(c) Describe what this approximate system tells about the rate of change of each of the specie populations x(t) and y(t) near (0, 0).
(d) Find the solution for the approximate system given in part (b).
(e) Sketch x (t) and y (t) as determined in part (d) on the same coordinate plane.
(f) Sketch a phase trajectory through (0.001; 0.002) for the predator-prey system. Describe in words what happens to each population of species and the interaction between them.
ANS:
(a) or
(b) Near ,
(c) Near , the prey population increases exponentially, and the predator population decreases exponentially.
(d)
(e)
(f)
As the population of predator decreases, the population of prey increases.
7. Consider the following predator-prey system where x and y are in millions of creatures and t represents time in years:
(a) Show that (4, 2) is the nonzero equilibrium solution.
(b) Find an expression for .
(c) The direction field for the differential equation is given below:
(i) Locate (4, 2) on the graph.
(ii) Sketch a rough phase trajectory through P indicated in the graph.
(d) With the aid of the phase trajectory, answer the following questions:
(i) For the region and 0 < y < 2, is x (t) increasing or decreasing? Is y (t) increasing or decreasing? Describe in words how the two species interact with one another.
(ii) For the region x > 4 and 0 < y < 2, is x (t) increasing or decreasing? Is y (t) increasing or decreasing? Describe in words how the two species interact with one another.
(iii) For the region x > 4 and y > 2, is x (t) increasing or decreasing? Is y (t) increasing or decreasing? Describe in words how the two species interact with one another.
(iv) For the region 0 < x < 4 and y > 2, is x (t) increasing or decreasing? Is y (t) increasing or decreasing? Describe in words how the two species interact with one another.
(e) Suggest a pair of species which might interact in the manner described by this system.
ANS:
(a) Solve the system of equations
(b)
(c)
(d) (i)
In this region the number of predator decreases because of the lack of prey, whereas the prey population can increase due to lack of predators.
(ii)
In this region the prey population has increased so much that the predator population can also increase.
(iii)
In this region the predator population has increased so much that the prey population is in decline.
(iv)
In this region, due to the lack of prey, both predator and prey are in decline.
(e) Wolves and rabbits.
8. A phase portrait of a predator-prey system is given below in which F represents the population of foxes (in thousands) and R the population of rabbits (in thousands).
(a) Referring to the graph, what is a reasonable non-zero equilibrium solution for the system?
(b) Write down a possible system of differential equations which could have been used to produce the given graph.
(c) Describe how each population changes as time passes, using the initial condition P indicated on the graph.
(d) Use your description in part (c) to make a rough sketch of the graph of R and F as functions of time.
ANS:
(a) (4, 2)
(b) (Answers may vary.)
(c) Initially, the numbers of both species increase. At a certain point, the rabbit population begins a steep decline, followed closely by the fox population. Then the rabbit population begins to increase, again followed by the fox population, and the cycle begins anew.
(d)
9. The population of two species is modeled by the system of equations .
(a) Find an expression for .
(b) A possible direction field for the differential equation in part (a) is given below:
Use this graph to sketch a phase portrait with each of P, Q, R, and S as an initial condition. Describe the behavior of the trajectories near the nonzero equilibrium solutions.
(c) Graph x and y as function of t. What happens to the population of the two species as the time t increases without bound?
ANS:
(a)
(b)
(c)
The predator and prey populations rise and fall in cycles.
10. In each of the given systems, x and y are populations of two different species which are solutions to the differential equations. For each system, describe how the species interact with one another (for example, do they compete for the same resources, or cooperate for mutual benefit?) and suggest a pair of species that might interact in a manner consistent with the given system of equations.
(a) (d)
(b) (e)
(c) (f)
ANS:
(a) Predator-prey system for example, robins and worms.
(b) Compete for the same resource for example, cheetahs and lions compete for wildebeest.
(c) Cooperate for mutual benefit for example, clownfish and anemone.
(d) Cooperate for mutual benefit for example, clownfish and anemone.
(e) Predator-prey system for example, whales and krills.
(f) No interaction for example, whales and tigers.
