376 Chapter 6: Differential Equations
6.3 Separation of Variables and the Logistic Equation
Multiple Choice
Identify the choice that best completes the statement or answers the question.
____ 1. Find the general solution of the differential equation .
a.
b.
c.
d.
e.
____ 2. Find the particular solution of the differential equation that satisfies the initial
condition .
a.
b.
c.
d.
e.
____ 3. Find an equation of the graph that passes through the point (7, 3) and has the slope
.
a.
b.
c.
6.3 Separation of Variables and the Logistic Equation 377
d.
e.
____ 4. Determine whether the function is homogeneous and
determine its degree if it is.
a. homogeneous, the degree is 4
b. homogeneous, the degree is 5
c. not homogeneous
d. homogeneous, the degree is 3
e. homogeneous, the degree is 2
____ 5. Sketch a few solutions of the differential equation on the slope field and then find the
general solution analytically.
a.
b.
c.
d.
e.
378 Chapter 6: Differential Equations
____ 6. A calf that weighs 70 pounds at birth gains weight at the rate where w
is weight in pounds and t is time in years. Use a computer algebra system to solve the differential
equation for
a.
b.
c.
d.
e.
____ 7. A calf that weighs 75 pounds at birth gains weight at the rate where w
is weight in pounds and t is time in years. If the animal is sold when its weight reaches 750 pounds, find
the time of sale using the model . Round your answer to two decimal places.
a. 8.17 years
b. 2.88 years
c. 0.92 year
d. 1 year
e. 0.32 year
____ 8. A calf that weighs 80 pounds at birth gains weight at the rate where w
is weight in pounds and t is time in years. What is the maximum weight of the animal if one uses the
model ?
a. 1500 lb
b. 880 lb
c. 1420 lb
d. 1580 lb
e. 2300 lb
____ 9. Find the orthogonal trajectories of the family .
a.
b.
c.
d.
e.
6.3 Separation of Variables and the Logistic Equation 379
____ 10. Find the orthogonal trajectories of the family
.
a.
b.
c.
d.
e.
____ 11. Identify the graph of the logistic function .
a. d.
b. e.
380 Chapter 6: Differential Equations
c.
____ 12. The logistic function models the growth of a population. Identify the
value of k.
a. 1.7
b. 2.2
c. 0.2
d. 20
e. 0.5
____ 13. The logistic function models the growth of a population. Identify the
maximum carrying capacity.
a. 20
b. 1.7
c. 0.5
d. 0.2
e. 2.2
____ 14. The logistic function models the growth of a population. Identify the
initial population.
a. 6
b. 8
c. 3
d. 24
e. 2
6.3 Separation of Variables and the Logistic Equation 381
____ 15. The logistic function models the growth of a population. Determine
when the population reaches one-half of the maximum carrying capacity. Round your answer to three
decimal places.
a. 0.549
b. 3.333
c. 1.151
d. 5.000
e. 1.000
____ 16. The logistic function models the growth of a population. Determine
when the population reaches of the maximum carrying capacity. Round your answer to three decimal
places.
a. 4.317
b. 3.000
c. 0.474
d. 0.677
e. 0.301
____ 17. Find the logistic equation that satisfies the following differential equation and initial
condition.
a.
b.
c.
d.
e. none of the above
382 Chapter 6: Differential Equations
____ 18. A conservation organization releases 40 coyotes into a preserve. After 4 years, there are
70 coyotes in the preserve. The preserve has a carrying capacity of 175. Write a logistic function that
models the population of coyotes in the preserve.
a.
b.
c.
d.
e.
____ 19. A conservation organization releases 50 foxes into a preserve. After 5 years, there are 85
foxes in the preserve. The preserve has a carrying capacity of 225. Determine the population after 10
years. Discard any fractional part of your answer.
a. 126
b. 118
c. 139
d. 131
e. 205
____ 20. A conservation organization releases 30 panthers into a preserve. After 3 years, there are
50 panthers in the preserve. The preserve has a carrying capacity of 150. Determine the time it takes for
the population to reach 110.
a. 13.139 years
b. 8.994 years
c. 10.378 years
d. 7.811 years
e. 12.003 years
6.3 Separation of Variables and the Logistic Equation 383
6.3 Separation of Variables and the Logistic Equation
Answer Section