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The logistic differential equation (dP/dt) = 0.09P(800 - P)

Question : The logistic differential equation (dP/dt) = 0.09P(800 - P) : 2163520

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

Solve the problem.

1) The logistic differential equation

(dP/dt) = 0.09P(800 - P)

describes the growth of a population P, where t is measured in years.

Find the carrying capacity of the population.

A) 7.2

B) 400

C) 800

D) 1600

2) The logistic differential equation

(dP/dt) = 0.0005P(1400 - P)

describes the growth of a population P, where t is measured in years.

Find the carrying capacity of the population.

A) 1400

B) 0.7

C) 700

D) 2800

3) The logistic differential equation

(dN/dt) = 0.044N(1000 - N)

describes the growth of a population N, where t is measured in years.

Find the intrinsic rate, r.

A) r = 0.044

B) r = 1000

C) r = 44,000

D) r = 44

4) The logistic differential equation

(dN/dt) = -0.02N2 + 2N

describes the growth of a population N, where t is measured in years.

Find the intrinsic rate, r.

A) r = 100

B) r = -0.02

C) r = 2

D) r = 0.02

SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.

5) Suppose an infectious disease spreads through an elementary school at a rate proportional to the product of the percentage of pupils who have the disease and the percentage of pupils who have not yet contracted the disease. Suppose that at the beginning of the epidemic 5% of the pupils have the disease. Let f(t) be the percentage of pupils who have the disease at time t; give the differential equation satisfied by f(t). Does the following accurately describe this situation: y' = ky(100 - y); y(0) = 0.05, where k is a positive constant?

Enter "yes" or "no".

6) A savings account earns 6% annual interest, compounded continuously. An initial deposit of $8500 is made, and thereafter money is withdrawn continuously at the rate of $480 per year. Does the following accurately represent this situation: y' = 0.06y - 480; y(0) = 8500?

Enter "yes" or "no".

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

7) A man opens a savings account that earns interest at an annual rate of 6% compounded continuously. He plans to make continuous withdrawals at a rate of $300 per year. What will happen if his initial deposit is $5000? [Hint: Let f(x) be the savings account balance at time t, and determine the differential equation satisfied by f(t).]

A) The balance will increase indefinitely.

B) The balance will remain at $5000 as long as the interest and withdrawals remain the same.

C) The balance will decrease until it runs out.

D) The balance will increase at an increasing rate until it reaches $18,000, at which point it will increase at a decreasing rate.

E) none of these

8) A population of algae consists of 4000 algae at time t = 0. Conditions will support at most 600,000 algae. Assume that the rate of growth of algae is proportional both to the number present (in thousands) and to the difference between 600,000 and the number present (in thousands). Write a differential equation using 0.03 for the constant of proportionality.

A) dy/dt = 0.03y(y - 600)

B) dy/dt = 4000y(600 - 0.03y)

C) dy/dt = 0.03y(600 - y)

D) dy/dt = 0.03(600 - y)

9) A certain developing country has a population of 500,000. The yearly rate of increase of literacy among the people is proportional to the number of illiterate people in the population. Letting f(t) represent the number of literate people, determine the differential equation that f(t) satisfies. (Let k represent a positive constant.)

A) f'(t) = 500,000 - kf(t)

B) f'(t) = k(500,000 - f(t))

C) f'(t) = (kf(t)/500,000)

D) f'(t) = 500,000(1 - f(t))

E) none of these

10) Suppose that a substance A is converted to substance B at a rate that is proportional to the cube of the amount of B present. The amount of A and B together is always constant, say M. If f(t) = y is the amount of A present at time t, then which of the following differential equation describes the situation?

A) y' = ky3; k < 0

B) y' = k(M - y)3; k > 0

C) y' = k(M - y)3; k < 0

D) y' = ky3; k > 0

E) none of these

SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.

11) A certain drug is introduced into a person's bloodstream. Suppose that the rate of decrease of the concentration of the drug in the blood is directly proportional to the product of two quantities: (a) the amount of time elapsed since the drug was introduced, and (b) the square of the concentration. Let y = f(t) denote the concentration of the drug in the blood at time t. Set up a differential equation satisfied by f(t). Does the following accurately describe this situation: y' = kty2, where k is a negative constant ?

Enter "yes" or "no".

12) A certain chemical vaporizes when exposed to the air. Suppose f(t) is the amount of chemical present. It is found that the rate of vaporization of the chemical is proportional to the amount of chemical present squared. Write a differential equation satisfied by f(t) . Does this equation accurately represent this situation: y' = ky2; k < 0 ?

Enter "yes" or "no".

13) Depending on the type of soil there is a constant M that represents the maximum amount of water the soil can absorb per cubic ft. If the rate of absorption is proportional to the difference between the maximum amount of water that could be absorbed and the amount of water that has been absorbed, write a differential equation satisfied by y = f(t), the amount of water in the soil at time t. Does this equation accurately describe this situation: y' = k(M - y); k > 0 ?

Enter "yes" or "no".

14) A sports enthusiast drinks 2liters of water per hour. Water is eliminated from the body at a rate proportional to the amount of water in the body (due to perspiration). Write a differential equation satisfied by f(t), the amount of water in the body. Does this equation accurately describe this situation: y' = 2 - ky ?

Enter "yes" or "no".

15) A patient is receiving a steady infusion of glucose. Let y denote the concentration of glucose in the blood at time t , measured in milligrams of glucose per 100 cubic centimeters of blood, and suppose that y satisfies the differential equation y' = 48 - 0.4y. What will be the approximate concentration of glucose in the blood after a long period of time, provided the glucose infusion is continued at the same rate?

Enter just an integer representing the number of mg glucose per 100cc blood (no units)

16) After a baby whale is born, its weight gain at any time is proportional to the product of its weight and the difference between its weight and its weight at maturity. Give a differential equation satisfied by f(t), its weight at time t. Does the following accurately describe this situation?

y' = ky(M - y);

M = weight at maturity;

k > 0

Enter "yes" or "no".

17) The birth rate in a certain city is 2% per year and the death rate is 2.5% per year. Also, there is a net movement of population into the city at the rate of 4000 people per year. Let N = f(t) be the city's population at time t. Write the differential equation satisfied by f(t). Does this equation accurately represent this situation: y' = 4000 - 0.005y?

Enter "yes" or "no".

18) An investment earns 25% interest per year. Every year $10,000 is withdrawn in order to pay dividends to the investors. Set up a differential equation satisfied by f(t), the amount of money invested at time t. Does this equation accurately describe this situation: y' = 0.25y - 10,000?

Enter "yes" or "no".

19) A millionaire wants to set up a trust for her grandchild. She wants to put a lump sum of money into an account earning 10% interest. She'd like her grandchild to be able to withdraw $100 every month for the rest of the child's life. Write a differential equation satisfied by f(t), the amount of money in the account at time t. Does the equation, y' = 0.1y - 100, accurately describe this situation?

Enter "yes" or "no".

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

20) Suppose that an epidemic is spreading at a rate proportional to the square of the infected population. Let f(t) be the number of infected people at time t, and suppose y = f(t) satisfies a differential equation y' = g(y). Which of the following sets of curves could represent solutions of y' = g(y)? [Hint: First determine the differential equation y' = g(y) .]

A)

B)

C)

D)

Solution
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