Question 1

Which of the following graphs best describes the temperature in a large North American city over the course of a year?

Accepted Answer

I

Question 2

A quantity y satisfies the differential equation  . Thus, y is decreasing when y is ________(less/greater) than _____.

Accepted Answer

Part A: less
Part B: 0

Question 3

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

dB / dt = -$7000

Question 4

A cup of green tea contains 32 mg of caffeine when you are using the tea leaves for the first time. A cup from the second brew contains 12 mg of caffeine, while a cup from the third brew contains only 4 mg of caffeine. Caffeine leaves the body at a continuous rate of about 17% per hour.
a) Write a differential equation for the amount, C, of caffeine in the body at time t hours after drinking the green tea.
b) Use the differential equation to find  at the start of the first hour (right after drinking the tea) for a cup from the first brew, and use your answer to estimate the change in caffeine in the body during the first hour.
c) Does the initial amount of caffeine in the body (whether from the first, second or third brew) change the differential equation?

Accepted Answer

The answer of A cup of green tea contains 32...

Question 5

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 6

A population of birds introduced onto an island without predators grows at a rate proportional to the size of the population. Write a differential equation for the size of the population, P, as a function of time. Is the constant of proportionality positive or negative?
A)  , positive
B)  , negative
C)  , negative
D)  , positive

Accepted Answer

The answer of A population of birds introduced onto an...

Question 7

A quantity T satisfies the differential equation  .
a) Is T increasing or decreasing when T = -5?
b) For what value of T is the rate of change of T equal to zero?

Accepted Answer

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

Question 8

A spherical raindrop evaporates at a rate proportional to its surface area. If V = volume of the raindrop and S = surface area, which of the following is a differential equation for  ?
A)  with k a negative constant
B)  with k a negative constant
C)  with k a positive constant
D)  with k a positive constant

Accepted Answer

The answer of A spherical raindrop evaporates at a rate...

Question 9

Which one(s) of the following are solutions to the differential equation  ?
A) 
B) 
C) 
D) 
E)

Accepted Answer

The answer of Which one(s) of the following are solutions...

Question 10

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

Accepted Answer

The answer of Carbon-14 decays at a rate proportional to...

Question 11

A bank account initially containing $4000 earns interest at a continuous rate of 6% per year. Deposits are made into the account at a constant rate of $500 per year. Which is the differential equation for the balance, B, in the account as a function of time, t, in years?
A) 
B) 
C) 
D)

Accepted Answer

The answer of A bank account initially containing $4000 earns...

Question 12

A population of rodents grows at a rate proportional to the size of the population. Which of the following is the the differential equation for the size of the population, P, as a function of time, t?
A)  , with k positive
B)  , with k negative
C)  , with k positive
D)  , with k negative

Accepted Answer

The answer of A population of rodents grows at a...

Question 13

A drug is administered intravenously to a patient at a rate of 12 mg per day. About 40% of the drug in the patient's body is metabolized and leaves the body each day. Which is the differential equation for the amount of the drug, D, in the body as a function of time, t, in days?
A) 
B) 
C) 
D)

Accepted Answer

Let D(t) be the amount of drug in the body at time t. Then the rate of change of D with respect to time is the difference between the amount of drug entering the body and the amount leaving due to metabolism. 
The amount of drug entering the body is constant at 12 mg per day. 
The amount leaving the body due to metabolism is 40% of the drug in the body, which is 0.4D(t). 
Therefore, the differential equation is:
dD/dt = 12 - 0.4D(t)
which can be rearranged as:
dD/dt + 0.4D(t) = 12
This is a first-order linear differential equation, which has the general solution:
D(t) = Ce^(-0.4t) + 30
where C is the constant of integration. 
Since the initial condition is not given, we cannot determine the value of C.

Question 14

There is a theory that says the rate at which information spreads by word of mouth is proportional to the product of the number of people who have heard the information and the number who have not. Suppose the total population is N. Which of the following differential equations describe the rate,  , at which the information spreads by word of mouth?
A) 
B) 
C) 
D)

Accepted Answer

The answer of There is a theory that says the...

Question 15

Water runs down a certain type of drainpipe at a rate proportional to the amount of water on the roof after a rainfall. Write a differential equation for the amount of water, W, on the roof at time t minutes after the rain stops.
A) 
B) 
C)  
D)

Accepted Answer

The answer of Water runs down a certain type of...

Question 16

Water is being pumped into a pool at a rate of 150 gallons per day, and is evaporating at a rate of 0.2% per day. Which is the differential equation for the amount, A, of gallons of water in the pool as a function of time, t, in days?
A) 
B) 
C) 
D)

Accepted Answer

Let's denote the amount of water in the pool by A. 
The rate at which water is being pumped into the pool is 150 gallons per day. 
The rate at which water is evaporating from the pool is 0.2% per day.

Therefore, the differential equation for the amount, A, of gallons of water in the pool as a function of time, t, in days is:

$\frac{dA}{dt} = 150 - 0.002A$

This is because the change in the amount of water in the pool (dA/dt) is equal to the inflow (150 gallons per day) minus the outflow (0.2% of the current amount of water in the pool, which is 0.002A).

Thus, the correct choice is D.

Question 17

A quantity Q satisfies the differential equation  . For what value of Q is the rate of change equal to 0?

Accepted Answer

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

Question 18

Which of the following graphs best describes the population of a species that is introduced to a confined space?
I.  II.  III.  IV.

Accepted Answer

The answer of Which of the following graphs best describes...

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

Accepted Answer

The answer of Consider the differential equation for the logistic...

Question 20

The deer population, P, in an area is increasing at a rate of 25% per year due to breeding. At the same time, about 200 deer are shot by hunters each year. Which is the differential equation for the population of deer as a function of time t, in years?
A) 
B) 
C) 
D)

Accepted Answer

The answer of The deer population, P, in an area...

Quiz 9: Mathematical Modeling Using Differential Equations

Which of the following graphs best describes the temperature in a large North American city over the course of a year?

A quantity y satisfies the differential equation . Thus, y is decreasing when y is ____(less/greater) than _.

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

A population of birds introduced onto an island without predators grows at a rate proportional to the size of the population. Write a differential equation for the size of the population, P, as a function of time. Is the constant of proportionality positive or negative?

A quantity T satisfies the differential equation . a) Is T increasing or decreasing when T = -5? b) For what value of T is the rate of change of T equal to zero?

A spherical raindrop evaporates at a rate proportional to its surface area. If V = volume of the raindrop and S = surface area, which of the following is a differential equation for ?

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

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?

A bank account initially containing $4000 earns interest at a continuous rate of 6% per year. Deposits are made into the account at a constant rate of $500 per year. Which is the differential equation for the balance, B, in the account as a function of time, t, in years?

A population of rodents grows at a rate proportional to the size of the population. Which of the following is the the differential equation for the size of the population, P, as a function of time, t?

A drug is administered intravenously to a patient at a rate of 12 mg per day. About 40% of the drug in the patient's body is metabolized and leaves the body each day. Which is the differential equation for the amount of the drug, D, in the body as a function of time, t, in days?

Water runs down a certain type of drainpipe at a rate proportional to the amount of water on the roof after a rainfall. Write a differential equation for the amount of water, W, on the roof at time t minutes after the rain stops.

Water is being pumped into a pool at a rate of 150 gallons per day, and is evaporating at a rate of 0.2% per day. Which is the differential equation for the amount, A, of gallons of water in the pool as a function of time, t, in days?

A quantity Q satisfies the differential equation . For what value of Q is the rate of change equal to 0?

Which of the following graphs best describes the population of a species that is introduced to a confined space? I. II. III. IV.

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

The deer population, P, in an area is increasing at a rate of 25% per year due to breeding. At the same time, about 200 deer are shot by hunters each year. Which is the differential equation for the population of deer as a function of time t, in years?