Deck 8: Mathematical Modeling With Differential Equations

Question 1

Solve the differential equation.

Answer 1

Question 2

What is the solution to the following differential equation?

A)
B)
C)
D)
E)

Answer 2

Question 3

Suppose , find the value of the slope field at .

A)
B)
C)
D)
E)

Answer 3

Question 4

Solve the differential equation.

Question 5

What is the solution to the following differential equation?

A)
B)
C)
D)
E)

Question 6

A tank is filled with gallons of water with oz of salt dissolved in it. At t = 0, salt solution containing oz/gal enters the tank at a rate of gal/min. Well-mixed solution leaves at the same rate. Write down an expression that gives the amount of salt S, in the tank at any time t.

Question 7

Use Euler's method with the step size of to approximate solution to the initial-value problem at .

A)
B)
C)
D)
E)

Question 8

Solve the following differential equation.

Question 9

Solve the following differential equation.

Question 10

What is the solution to the following differential equation?

A)
B)
C)
D)
E)

Question 11

Solve the following differential equation.

Question 12

Suppose , find the value of the slope field at .

A)
B)
C)
D)
E)

Question 13

A tank is filled with gallons of water in which oz of salt is dissolved. At t = 0, salt solution containing oz/gal enters the tank at a rate of gal/min. Well mixed solution leaves at the same rate. Write down an expression that gives the amount of salt S, in the tank at any time t.

Question 14

Suppose , find the value of the slope field at .

A)
B)
C)
D)
E)

Question 15

Suppose , find the value of the slope field at .

A)
B)
C)
D)
E)

Question 16

Solve the differential equation by separation of variables.

Question 17

Suppose . Find the value of the slope field at .

A)
B)
C)
D)
E)

Question 18

Solve the following initial-value problem.

Question 19

A tank initially contains gal of pure water. At time a solution containing oz of dissolved salt per gal flows into the tank at gal/min. The well stirred mixture is pumped out of the tank at the same rate. Write down an expression for the amount of salt in the tank at any time. How much salt is present at the end of minutes?

Question 20

Solve the differential equation using any method you wish.

Question 21

Use Euler's method with the step size of to compute the approximate solution to the initial-value problem at .

Question 22

What is the solution to the following differential equation?

A)
B)
C)
D)
E)

Question 23

Use Euler's method with the step size of to compute the approximate solution to the initial-value problem at . Round your answer to six decimal places if need be.

Question 24

Solve the differential equation using any method you wish.

Question 25

What is the solution to the following differential equation?

A)
B)
C)
D)
E)

Question 26

Use Euler's method with the step size of to approximate solution to the initial-value problem at .

A)
B)
C)
D)
E)

Question 27

Use Euler's method with the step size of to compute the approximate solution to the initial-valued problem at . Round your answer to six decimal places if need be.

Question 28

What is the solution to the following differential equation?

A)
B)
C)
D)
E)

Question 29

What is the solution to the following differential equation?

A)
B)
C)
D)
E)

Question 30

Use Euler's method with the step size of to approximate solution to the initial-value problem at .

A)
B)
C)
D)
E)

Question 31

What is the solution to the following initial-value problem? (Use separation of variables!)

A)
B)
C)
D)
E)

Question 32

What is the solution to the following differential equation?

Question 33

Use Euler's method with the step size of to approximate solution to the initial-value problem at . You will need to take two steps here!

A)
B)
C)
D)
E)

Question 34

Solve the differential equation by separation of variables.

Question 35

Use Euler's method with the step size of to approximate solution to the initial-value problem at .

A)
B)
C)
D)
E)

Question 36

What is the solution to the following differential equation? (Use separation of variables!)

A)
B)
C)
D)
E)

Question 37

Solve the differential equation by separation of variables.

Question 38

Use Euler's method with the step size of to approximate solution to the initial-value problem at .

A)
B)
C)
D)
E)

Question 39

What is the solution to the following differential equation? (Use separation of variables!)

A)
B)
C)
D)
E)

Question 40

Solve the differential equation by separation of variables.

Question 41

Solve the initial-value problem.

Question 42

Find the exponential decay model that satisfies , if the halving time is years. Round any numerical quantities to 6 decimal places if need be.

A)
B)
C)
D)
E)

Question 43

Find the exponential decay model that satisfies , if the halving time is years.

A)
B)
C)
D)
E)

Question 44

Solve the initial-value problem.

Question 45

For radioactive carbon-14, If 330 g of carbon exist today in a sealed container, how many grams of carbon-14 will be left after years? Round your answer to 6 decimal places.

Question 46

A particle moving along the x-axis encounters a resisting force that results in an acceleration of Given that cm and v = cm/s at t = 0, find the velocity v as a function of t.

Question 47

If the situation modeled is

A) increasing
B) decreasing
C) remaining constant
D) More information is needed.
E) Cannot be determined.

Question 48

Suppose that a radioactive substance decays with a half-life of years. Find a function that expresses the amount of substance present at time t if there are g of the substance initially present. Round answers to 3 decimal places if need be.

A)
B)
C)
D)
E)

Question 49

A particle moving along the x-axis encounters a resisting force that results in an acceleration of Given that cm and v = 35 cm/s at t = 0, find the position x as a function of t.

Question 50

Solve the initial-value problem.

Question 51

Find the exponential growth model that satisfies , if the doubling time is years.

A)
B)
C)
D)
E)

Question 52

A cup of water with a temperature of C is placed in a room at constant temperature C. It takes minutes for the cup to cool to . Assuming Newton's Law of Cooling applies, find the temperature, T, of the cup of water at any time t. Express all numerical quantities in exact form.

Question 53

Solve the initial-value problem.

Question 54

Find the exponential growth model that satisfies , if the doubling time is years. Round any numerical quantities to 6 decimal places if need be.

A)
B)
C)
D)
E)

Question 55

Solve the initial-value problem.

Question 56

If the situation modeled is

A) increasing
B) decreasing
C) remaining constant
D) More information is needed.
E) Cannot be determined.

Question 57

If the situation modeled is

A) increasing
B) decreasing
C) remaining constant
D) More information is needed.
E) Cannot be determined.

Question 58

A cup of water with a temperature of C is placed in a room with constant temperature C. It takes minutes for the cup to cool to . Assuming Newton's Law of Cooling applies, find the temperature, T, of the cup of water at any time t. Express all numerical quantities in exact form.

Question 59

A certain radioactive substance has a half-life of years. Approximately how many years will it take so that only of the original amount remains? Round your answers to six decimal places.

Question 60

Suppose 5 grams of a radioactive substance decays according to the equation If it takes years for the substance to reduce to of its original mass, write an equation for the amount of the substance present as a function of time.

Question 61

State the order of the differential equation, and decide which family of functions is a solution.

A) ; 2nd order
B) ; 1st order
C) ; 1st order
D) ; 2nd order
E) ; 2nd order

Question 62

Solve the initial-value problem.

Question 63

State the order of the differential equation, and decide which family of functions is a solution.

A) ; 2nd order
B) ; 1st order
C) ; 1st order
D) ; 2nd order
E) ; 2nd order

Question 64

Solve the initial-value problem.

Question 65

If the situation modeled is

A) increasing
B) decreasing
C) remaining constant
D) More information is needed.
E) Cannot be determined.

Question 66

Solve the initial-value problem.

Question 67

Solve the initial-value problem.

Question 68

Suppose that an initial population of bacteria dies exponentially at a rate of % per hour. How many bacteria are present after t hours?

A)
B)
C)
D)
E)

Question 69

Suppose that an initial population of bacteria grows exponentially at a rate of % per hour. How many bacteria are present after t hours?

A)
B)
C)
D)
E)

Question 70

Solve the initial-value problem.

Question 71

State the order of the differential equation, and decide which family of functions is a solution.

A) ; 1st order
B) ; 1st order
C) ; 2nd order
D) ; 2nd order
E) ; 2nd order

Question 72

If the situation modeled is

A) increasing
B) decreasing
C) remaining constant
D) More information is needed.
E) Cannot be determined.

Question 73

If the situation modeled is

A) increasing
B) decreasing
C) remaining constant
D) More information is needed.
E) Cannot be determined.

Question 74

Solve the initial-value problem.

Question 75

State the order of the differential equation, and decide which family of functions is a solution.

A) ; 1st order
B) ; 1st order
C) ; 2nd order
D) ; 1st order
E) ; 2nd order

Question 76

Solve the initial-value problem.

Question 77

A 100-kg block stretches a spring 0.05 m from its initial position. If the spring is then stretched 0.2 m and released, write an equation for its position relative to its equilibrium position.