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Exam 9: Differential Equations

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Question 1

Multiple Choice

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Solve the problem. -A 1000-gallon holding tank contains 100 gallons of water. Initially, each gallon of water in the tank contains 3 pounds of pollutants. Water containing 4 pounds of pollutants per gallon enters the tank at a rate of 50 gallons Per hour, and the uniformly mixed water is released from the tank at a rate of 25 gallons per hour. How many Pounds of pollutants are in the tank after 4 hours? At what rate (in pounds per gallon) are the pollutants being Released after 4 hours?

A) There are 250 pounds of pollutants in the tank after 4 hours. The pollutants are being released at a rate of 1.25 pounds per gallon after 4 hours.
B) There are 650 pounds of pollutants in the tank after 4 hours. The pollutants are being released at a rate of 5.75 pounds per gallon after 4 hours.
C) There are 180 pounds of pollutants in the tank after 4 hours. The pollutants are being released at a rate of 4.35 pounds per gallon after 4 hours.
D) There are 750 pounds of pollutants in the tank after 4 hours. The pollutants are being released at a rate of 3.75 pounds per gallon after 4 hours.

Correct Answer

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D

Question 2

Multiple Choice

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Replace the ? with a function of x that will make the integrand equal to the derivative of a product, and then find the antiderivative. Choose 0 for the constant of integration. -

A)
B)
C)
D)

Correct Answer

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C

Question 3

Multiple Choice

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Replace the ? with a function of x that will make the integrand equal to the derivative of a product, and then find the antiderivative. Choose 0 for the constant of integration. -

Find the general solution for the differential equation. Do not attempt to find an explicit expression for the solution. -

Use the given slope field to graph the particular solutions obtained by letting C = -2, -1, 0, 1, and 2 for the general solution shown. -

Find the general solution for the differential equation. Then find the particular solution satisfying the initial condition. -

Find the integrating factor I(x) for each equation, and then find the general solution. -

Solve the problem. -A company wishes to analyze a new room deodorizer. The active ingredient evaporates at a rate proportional to the amount present. One third of the ingredient evaporates in the first 20 days after the deodorizer is installed. If The deodorizer becomes ineffective after 85% of the active ingredient has evaporated, how long will one of these Deodorizers remain effective? Round to the nearest whole number as needed.

Replace the ? with a function of x that will make the integrand equal to the derivative of a product, and then find the antiderivative. Choose 0 for the constant of integration. -

Find the general solution for the differential equation. Then find the particular solution satisfying the initial condition. -

Find the integrating factor, the general solution, and the particular solution satisfying the given initial condition. -

Use the given slope field to graph the particular solutions obtained by letting C = -2, -1, 0, 1, and 2 for the general solution shown. -

Solve the problem. -A culture of bacteria is growing in a medium that can support a maximum of 1,500 bacteria. The rate of change of the number of bacteria is proportional to the product of the number present and the difference between 1,500 and the Number present. The culture initially contains 250 bacteria. After 1 hour, there are 350 bacteria. a. How many bacteria are present after 3 hours? Round to the nearest whole number as needed. b. When will the culture contain 1,250 bacteria? Round to the nearest tenth as needed.

Show that the technique of separation of variables is applicable by writing the differential equation in the form = g(x) . -

Solve the problem. -The price p(t) of a produce is assumed to satisfy the differential equation (A) Show that is the general solution of this differential equation, and evaluate (B) Graph the particular solutions that satisfy the initial conditions p(0) = 3 and p(0) = 12. (C) Discuss the long-term behavior of the price of this product.

Find the integrating factor I(x) for each equation, and then find the general solution. -

Use the given slope field to graph the particular solutions obtained by letting C = -2, -1, 0, 1, and 2 for the general solution shown. -

Find an explicit expression for the particular solution for the differential equation. -

Is the differential equation first-order linear? If so, find f(x) and g(x) so that the equation can be expressed in standard + f(x) y = g(x) . -

Use the given slope field to graph the particular solutions obtained by letting C = -2, -1, 0, 1, and 2 for the general solution shown. -