Question 1

Which one(s) of the following are solutions to the differential equation  ?

Accepted Answer

A)   
B)   
C)   
D)   
E)    D

Question 2

The amount of medicine present in the blood of a patient decreases due to metabolism according to the exponential decay model. One hour after a dose was given, there were 3.7 nanograms/cm³ present, and a hour later there were 2.5 ng/cm³. After how many hours will there be less than 0.8 ng/cm³ present, assuming no more medication is taken? Round to 1 decimal place.

Accepted Answer

4.9

Question 3

Mark all of the differential equations that are NOT separable.

Accepted Answer

A)   
B)   
C)   
D)    A, B, C

Question 4

A radioactive isotope decays at a continuous rate of approximately 15% per day. If A is the amount of the isotope and t is time in days, what is the differential equation for this situation and its general solution?

Accepted Answer

A)    ; 
B)    ; 
C)    ; 
D)    ;

Question 5

The fox eats the hare, so it needs it to survive. The hare would do fine without the fox. Which (if any) of the following systems of differential equations model the interaction between the fox and the hare, with the fox as x and the hare as y? Select all that apply.

Accepted Answer

A)    , 
B)    , 
C)    , 
D)  None of these

Question 6

A person withdraws money from a trust fund at a rate of $12,000 per year, and the account is earning interest at a rate of 5% per year, compounded continuously. Write a differential equation for the balance, B, in the account as a function of time, t, in years and use it to calculate  if B=$100,000.

Accepted Answer

The answer of A person withdraws money from a trust...

Question 7

A person receives a drug intravenously at the rate of 3 mg per hour. The drug is eliminated from the body at a rate proportional to the amount present with a constant of proportionality of k = -0.4. What is the long term amount of the drug in the body, once the system has stabilized?

Accepted Answer

The answer of A person receives a drug intravenously at...

Question 8

The equilibrium solution for  is P = _____. This solution is ________ (stable/unstable).

Accepted Answer

The answer of The equilibrium solution for  is P...

Question 9

A quantity Q satisfies the differential equation  . Is Q increasing or decreasing at Q = 3?

Accepted Answer

The answer of A quantity Q satisfies the differential equation...

Question 10

According to Newton, the rate at which the temperature of water in a swimming pool changes is directly proportional to the difference between the temperature L outside and the temperature T of the water in the pool. Suppose the temperature outside stays at a constant 28  C for two hours, and that during that time the temperature of the water increases from 20  C to 24  C. Which of the following shows the differential equation for this situation and its solution?

Accepted Answer

A)    ; 
B)    ; 
C)    ; 
D)    ;

Question 11

For S and I satisfying the differential equations   ,
is I increasing or decreasing when  ?

Accepted Answer

The answer of For S and I satisfying the differential...

Question 12

Given that  and that  , estimate  to 2 decimal places by first estimating y(1). Assume that the rate of growth given by  is approximately constant over each unit time interval.

Accepted Answer

The answer of Given that  and that  ,...

Question 13

A lake with constant volume V, in km³, contains a quantity of Q km³pollutant. Clean water enters the lake and causes a total outflow of r km³ per year. The rate at which the pollutant decreases at any time t equals the product of the pollutant Q per volume V and the rate at which the water flows out of the lake. If km³ and km³ per year, how many years will it take for the pollutant to decrease to half of its original quantity? Round to the nearest whole year.

Accepted Answer

The answer of A lake with constant volume V, in...

Question 14

According to Newton, the rate at which the temperature of water in a swimming pool changes is directly proportional to the difference between the temperature L outside and the temperature T of the water in the pool. Suppose the temperature outside stays at a constant 28  C for two hours, and that during that time the temperature of the water increases from 20  C to 24  C. What is the equilibrium solution to the equation modeling this situation?

Accepted Answer

The answer of According to Newton, the rate at which...

Question 15

Carbon-14 decays at a rate proportional to the amount present. Which of the following is the differential equation for the amount, C, of carbon-14 present at time t?

Accepted Answer

A)    , with k positive
B)    , with k negative
C)    , with k positive
D)    , with k negative

Question 16

If no more pollutants are dumped into a lake, the amount of pollution in the lake will decrease at a rate proportional to the amount of pollution present. If there are 400 units of pollution present initially and 184 units left after 8 years, use differential equations to find the number of units left after 13 years. Round to 1 decimal place.

Accepted Answer

The answer of If no more pollutants are dumped into...

Question 17

On January 1, 1879, records show that 500 of a fish called Atlantic striped bass were introduced into the San Francisco Bay. In 1899, the first year fishing for bass was allowed, 100,000 of these bass were caught, representing 10% of the population at the start of 1899. Owing to reproduction, at any time the bass population is growing at a rate proportional to the population at that moment. Assume that when fishing starts in 1899, the rate at which bass are caught is proportional to the square of the population with constant of proportionality  . What happens to the bass population in the long run?

Accepted Answer

A)  It approaches 0.
B)  It grows without bound.
C)  It approaches  .
D)  It approaches  .

Question 18

What is the solution to the differential equation  , given that P = 5 when t = 0?

Accepted Answer

The answer of What is the solution to the differential...

Question 19

Consider the differential equation for the logistic model representing a population of tarantulas introduced into a new habitat:  . What is the carrying capacity?

Accepted Answer

A)  200 tarantulas
B)  k tarantulas
C)  200 - P tarantulas
D)  1000 tarantulas

Question 20

For  to be a solution to the differential equation  , what must be the constant k?

Accepted Answer

A)  2
B)  0
C)  4
D)  k can be any number

Exam 9: Mathematical Modeling Using Differential Equations

Which one(s) of the following are solutions to the differential equation ?

Mark all of the differential equations that are NOT separable.

A radioactive isotope decays at a continuous rate of approximately 15% per day. If A is the amount of the isotope and t is time in days, what is the differential equation for this situation and its general solution?

The fox eats the hare, so it needs it to survive. The hare would do fine without the fox. Which (if any) of the following systems of differential equations model the interaction between the fox and the hare, with the fox as x and the hare as y? Select all that apply.

A person withdraws money from a trust fund at a rate of $12,000 per year, and the account is earning interest at a rate of 5% per year, compounded continuously. Write a differential equation for the balance, B, in the account as a function of time, t, in years and use it to calculate if B=$100,000.

A person receives a drug intravenously at the rate of 3 mg per hour. The drug is eliminated from the body at a rate proportional to the amount present with a constant of proportionality of k = -0.4. What is the long term amount of the drug in the body, once the system has stabilized?

The equilibrium solution for is P = _. This solution is ____ (stable/unstable).

A quantity Q satisfies the differential equation . Is Q increasing or decreasing at Q = 3?

For S and I satisfying the differential equations , is I increasing or decreasing when ?

Given that and that , estimate to 2 decimal places by first estimating y(1). Assume that the rate of growth given by is approximately constant over each unit time interval.

Carbon-14 decays at a rate proportional to the amount present. Which of the following is the differential equation for the amount, C, of carbon-14 present at time t?

What is the solution to the differential equation , given that P = 5 when t = 0?

Consider the differential equation for the logistic model representing a population of tarantulas introduced into a new habitat: . What is the carrying capacity?

For to be a solution to the differential equation , what must be the constant k?