Deck 9: Mathematical Modeling Using Differential Equations

Question 1

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

Answer 1

I

Question 2

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

Answer 2

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.

Answer 3

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?

Question 5

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

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

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?

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

Question 9

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

A)
B)
C)
D)
E)

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

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)

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

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)

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)

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)

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)

Question 17

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

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.

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

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)

Question 21

What is the solution of when ?

A)
B)
C)
D)

Question 22

If is a solution to the differential equation and y = 20 when t = 0, then k = _____ and C = _____.

Question 23

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

A)
B)
C)
D)
E)

Question 24

Is a solution to the differential equation ?

Question 25

Is the general solution to the differential equation ?

Question 26

Consider the differential equation . Is the general solution to the differential equation? If it is not, answer "not the solution". If it is the general solution and if y(3) = 5, what is the constant C?

A)yes, C = 121
B)yes, 0.30
C)yes, 12.23
D)not the solution

Question 27

If is a solution to the differential equation , then k = _____.

Question 28

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

A)
B)
C)
D)
E)

Question 29

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

A)
B)
C)
D)
E)

Question 30

Find the solution of the differential equation satisfying .

Question 31

Fill in the missing values in the table, given that . Assume the growth rate, given by , is approximately constant over each time interval.

Question 32

In Kenya, the population P for the recent past has obeyed the growth model , with t the number of years since 1990. The solution to the differential equation is of the form . If the population in 1990 was 24.64 million and in 1992 was 26.16 million, then A = _____ and k = _____. Thus, in the year 1997, the population was approximately _____ million. Round all answers to 2 decimal places.

Question 33

What is the solution of when ?

A)
B)
C)
D)

Question 34

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.

Question 35

Find the particular solution to the differential equation when .

A)
B)
C)
D)

Question 36

Does satisfy ?

Question 37

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

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

Question 38

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

Question 39

Is a solution to the differential equation ?

Question 40

Suppose satisfies the differential equation . Then k = _____ and C = _____. If either cannot be determined from the information given, enter "cannot tell".

Question 41

If the slope field for has constant slopes where x is fixed, what do we know about ?

A) depends only on y.
B) depends only on x.
C) must be a constant.
D)We can't determine anything about .

Question 42

Which of the following give a solution to the differential equation ? Select all that apply. first one: second one:

A)first one
B)second one
C)neither

Question 43

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.

Question 44

The following figure could be the slope field for the differential equation for and .

Question 45

Which of the following equations corresponds with the slope field shown below?
I. II. III. IV. None of them

Question 46

An anti-inflammatory drug has a half-life in the human body of about 8 hours.
A. Use the half-life to find the value of k in the differential equation , where Q is the quantity of the drug in the body t hours after the drug is administered. Round to 4 decimal places.
B. After how many hours will 45% of the original dose remain in the body? Round to 2 decimal places.

Question 47

Which of the following differential equations goes with the slope field in the figure?

A)
B)
C)
D)

Question 48

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?

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

Question 49

The following figure shows the slope field for the differential equation . Guess the equation of the solution curve that goes through the point (0,2).

Question 50

Money in a bank account grows continuously at an annual rate of 4%. Suppose $10,000 is put into an account at time t = 0. If B is the balance in the account after t years, what is the differential equation for B and its solution?

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

Question 51

Which of the following slope fields goes with the differential equation y?

Question 52

Look at the slope field labeled I. Consider a solution curve for slope field (I). Which of the answer choices describes the long-run behavior of y at various starting points?

A)No matter what the starting point, as , y approaches the value 4.
B)No matter what the starting point, as , y oscillates within a certain finite range.
C)Depending on the starting point, as , y approaches
.
D)Depending on the starting point, as , y approaches
or 0.

Question 53

The solution to the differential equation subject to the initial condition is , where k = _____ and C = _____. Round answers to 2 decimal places.

Question 54

For the first week, the spread of a rumor is proportional to the number of people who have heard the rumor. Find the particular solution to the differential equation for N, the number of people who have heard the rumor as a function of time in days, t, if 15 people have heard it at time t = 0, and 210 people have heard it at time t = 5.

A)
B)
C)
D)

Question 55

Sketch a slope field for the differential equation using the points indicated on the axes.

Question 56

On the slope field for the differential equation , sketch the solution curve in the fourth quadrant that goes through the point (0, -1).

Question 57

Consider the slope field for . What is the slope at the point (0,0)?

Question 58

Find a solution to the differential equation subject to the initial condition Q = 60 when t = 0.

Question 59

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

Question 60

The solution to the differential equation subject to the initial condition that y = 80 when x = 0 is , where k = _____ and C = _____.

Question 61

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 moment in time the bass population is growing at a rate proportional to the population at that moment. Write a differential equation satisfied by B(t), the number of Atlantic striped bass a time t, where t is in years since January 1, 1879 and 0 $\le$ t < 20 and solve it for B(t).

A)
B)
C)
D)

Question 62

Newton's Law of Cooling states that the rate of change of temperature of an object is proportional to the difference between the temperature of the object and the temperature of the surrounding air. A detective discovers a corpse in an abandoned building, and finds its temperature to be 24°C. An hour later its temperature is 16°C. Assume that the air temperature is 8°C, that normal body temperature is 37°C, and that Newton's Law of Cooling applies to the corpse. How many hours has the corpse been dead at the moment it is discovered? Round to 2 decimal places.

Question 63

What is the solution of the differential equation if when ?

A)
B)
C)
D)

Question 64

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?

Question 65

The logistic model for a population's growth is , where P is the size of the population in millions at any time t, measured in years. What is the size of the population when the rate of increase starts to decrease?

Question 66

You invest $2,500 in your nephew's catering business. He guarantees you a minimum return at a continuous interest rate of 4%. Of course, if the business continues to thrive, you will earn at a higher rate.
a) Write a differential equation for the minimum amount, B, of your return on investment at time t.
b) Solve the differential equation.
c) Graph the solution.

Question 67

What is the general solution of ?

Question 68

What is the general solution to the differential equation ?

A)
B)
C)
D)
E)none of the above

Question 69

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?

Question 70

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?

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

Question 71

A lake with constant volume V, in km³, contains a quantity of Q km³ pollutant. 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.

Question 72

A country experiences a continuous inflation rate of about 5.7% per year. If a t-shirt had a value of $12 in 1995, write a differential equation and use it to find what the t-shirt's value was in 2005. Round to the nearest cent.

Question 73

A certain bank account earns interest at the rate of 5% compounded continuously. Money is being withdrawn from the account in a continuous stream at a constant rate of $100,000 per year. Write a differential equation modeling how the balance B changes over time. Which of the following is the general solution, given an initial balance of B₀?

A)
B)
C)
D)

Question 74

A certain bank account earns interest at the rate of 5% compounded continuously. Money is being withdrawn from the account in a continuous stream at a constant rate of $100,000 per year. Use differential equations to determine what the minimum initial balance should be in order for the account never to be depleted.

Question 75

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.

Question 76

What is the general solution of ?

Question 77

A company earns a continuous annual rate of 11% of its net worth. At the same time, it has expenses of 6.2 million dollars per year. If the company's net worth at time t = 0 is 50 million, how many years will it take to go bankrupt? Round to the nearest year.

Question 78

A company earns a continuous annual rate of 11% of its net worth. At the same time, it has expenses of 6.4 million dollars per year. Write a differential equation for the company's worth, W, in millions of dollars as a function of time t, in years. What is the general solution to your differential equation?

A)
B)
C)
D)

Question 79

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

Question 80

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