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Question : Rats are not native to the islands off the western coast of South America : 2151685

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.

Solve the problem.

1) Rats are not native to the islands off the western coast of South America. However, rats are often introduced accidentally to an island by visiting ships. The population of introduced rats follows the logistic function with k = 0.00024 and t in months. Assume that there are 6 rats initially and that the maximum population size is 11,000. Find the rate of growth of the population after 6 months.

A) 26 rats/month

B) -2 rats/month

C) 134 rats/month

D) 7 rats/month

2) The following formula accurately models the relationship between the size of a certain type of tumor and the amount of time that it has been growing:

V(t) = 400(1 - e^{-0.0019t})^{3},

where t is in months and V(t) is measured in cubic centimeters. Calculate the rate of change of tumor volume at 160 months.

A) 0.085 cm^{3}/month

B) 0.116 cm^{3}/month

C) 0.501 cm^{3}/month

D) 0.181 cm^{3}/month

3) Researchers have found that the maximum number of successful trials that a laboratory rat can complete in a week is given by

P(t) = 50(1 - e^{-0.4t}),

where t is the number of weeks the rat has been trained. What is the maximum number of successful trials that a laboratory rat can complete in a week after being trained for 3 weeks.

A) 16

B) 6

C) 65

D) 35

4) Researchers have found that the maximum number of successful trials that a laboratory rat can complete in a week is given by

P(t) = 59(1 - e^{-0.4t}),

where t is the number of weeks the rat has been trained. Find the rate of change P'(t).

A) P'(t) = 59e^{-0.4t}

B) P'(t) = -23.6e^{-0.4t}

C) P'(t) = 23.6e^{-0.4t}

D) P'(t) = 59(1 + 0.4e^{-0.4t})

5) The natural resources of an island limit the growth of the population to a limiting value of 4302. The population of the island is given by the logistic equation

P(t) = (4302/1 + 4.63e^{-0.3t}),

where t is the number of years after 1980. What is the population of the island in 1987?

A) 2471

B) 971

C) 2608

D) 2745

6) The natural resources of an island limit the growth of the population to a limiting value of 3172. The population of the island is given by the logistic equation

P(t) = (3172/1 + 5.95e^{-0.38t}),

where t is the number of years after 1980. Find the rate of change P'(t).

A) P'(t) = (3172 - 7171.9e^{-0.38t}/(1 + 5.95e^{-0.38t})^{2})

B) P'(t) = (18,873e^{-0.38t}/(1 + 5.95e^{-0.38t})^{2})

C) P'(t) = (7171.9e^{-0.38t}/1 + 5.95e^{-0.38t})

D) P'(t) = (7171.9e^{-0.38t}/(1 + 5.95e^{-0.38t})^{2})

Provide an appropriate response.

7) If Q = 51e^{-0.7t} what happens to Q and to Q' as t increases?

A) Q decreases and Q' decreases.

B) Q increases and Q' increases.

C) Q decreases and Q' increases.

D) Q increases and Q' decreases.

8) If Q = 66e^{0.7t} what happens to Q and to Q' as t increases?

A) Q increases and Q' increases.

B) Q decreases and Q' decreases.

C) Q decreases and Q' increases.

D) Q increases and Q' decreases.

9) If Q = 72 - e^{-0.2t} what happens to Q and to Q' as t increases?

A) Q increases and Q' increases.

B) Q decreases and Q' decreases.

C) Q increases and Q' decreases.

D) Q decreases and Q' increases.

10) If Q = 109e^{0.5t} what happens to Q and to Q' as t increases?

A) Q decreases and Q' decreases.

B) Q increases and Q' decreases.

C) Q increases and Q' increases.

D) Q decreases and Q' increases.

11) Let A(t) represent a quantity which is growing exponentially. The percentage rate of growth (A'(t)/A(t)) is ____.

A) Increasing

B) Decreasing

C) Constant

D) None of these

12) A quantity Q is increasing by 4200 per year at the present time. This means that ____. (Provide a statement that is always true, involving either Q'(0), Q(0), Q(4200), or Q'(4200).)

A) Q'(0) = 4200

B) Q(4200) = 0

C) Q(0) = 4200

D) Q'(4200) = 0

13) A(t) = P(1 + r)nt represents the amount of money in an account paying interest compounded n times per year. The percentage rate of growth (A'(t)/A(t)) is ____.

A) Decreasing

B) Increasing

C) Constant

D) None of these

14) The derivative of 7^{x} =

A) (7^{x}/ln7)

B) 7^{x}ln7

C) x^{7x - 1}

D) 7^{x}