Deck 4: Exponential Functions

Question 1

An investment grows according to the formula . How many years will it take for the original investment to quadruple? Round to 1 decimal place.

Answer 1

34.7

Question 2

The price of an item increases due to inflation. Let give the price of the item as a function of time in years, with t = 0 in 2004. At what continuous annual rate is the price increasing? Round to 2 decimal places.

Answer 2

3.54%

Question 3

Is the formula for a function representing a quantity which begins at N in year t = 0 and grows at a constant annual rate of r% given by ?

Answer 3

False

Question 4

Is the formula for a function representing a quantity which begins at 2N in year t = 0 and grows at a continuous annual rate of r% given by ?

Question 5

Is the formula for a function representing a quantity which begins at an amount 35% larger than N in year t = 0 and grows at a continuous annual rate of r% given by
?

Question 6

Is the formula for a function representing a quantity which begins at N in year t = 0 and grows at a continuous annual rate of % given by ?

Question 7

Let give the size of a population of animals in year t. What will the population be after 19 years? Round to the nearest whole number.

Question 8

Let give the size of a population of animals in year t. After how many years will the population be approximately 770?

Question 9

The US population in 2005 was approximately 296.4 million. Assume the population increases at a rate of 1.32% per year. What is the formula for
, the population for the United States t years after 2005?

A)
B)
C)
D)

Question 10

The figure below shows the graphs of the following functions:
(A)

(B)

(C)

(D)

Which one is the graph of D?

Question 11

?

Question 12

?

Question 13

?

Question 14

Which of the following is correctly ordered from least to greatest?

A)
B)
C)
D)
E)
F)

Question 15

Jeff has $^20,000.He placed $10,000 in Bank 1 with an account earning 3.1% annual interest, compounded continuously. He also placed $10,000 in Bank 2 with an account earning 3.7% annual interest compounded weekly. After 10 years,

A) how much money is in Bank 1?
B) how much money is in Bank 2?

Question 16

You invest $6000 in an account earning 2.5% annual interest, compounded continuously. After how many years is the investment worth $14,000? Round your answer to two decimal places, if necessary.

Question 17

?

Question 18

An ant population grows at a continuous growth rate of 11.2%. If the population starts with 27,000 ants, how many ants are there after 6 months? Round your answer to the nearest ant.

Question 19

Without a calculator, graph the following on the same set of axes:

A)
B)
C)

Question 20

Which of the following quantities
Are increasing with time t ?

A)
B)
C)
D)

Question 21

For an account paying 9% annual interest, compounded continuously, the nominal rate is _____ % and the effective rate is _____ %. Round to 3 decimal places.

Question 22

What is the balance after 10 years in an account containing $600 that earns 4% interest compounded monthly? Round to the nearest cent.

Question 23

Which bank has the best effective annual yield?

A)Bank 1 with a nominal rate of 6.45% compounded monthly.
B)Bank 2 with a nominal rate of 6.33% compounded weekly.
C)Bank 3 with a nominal rate of 6.6% compounded yearly.

Question 24

What is the maximum effective annual yield among the following three banks? Give your answer correct to four decimal places.

A)Bank 1 with a nominal rate of 6.5% compounded monthly.
B)Bank 2 with a nominal rate of 6.35% compounded weekly.
C)Bank 3 with a nominal rate of 6.55% compounded yearly.

Question 25

You place $10,000 in an account. You hope to have $22,000 in the account after 15 years. What effective annual yield is needed to accomplish this? Give your answer correct to four decimal places.

Question 26

Suppose you would like to have $20,000 in 10 years. What is the minimum amount you need to deposit into a bank account earning 3% compounded weekly to reach this goal?

Question 27

Use the formula to answer the following questions about the investment it describes. Units are dollars and years.

A) Is the investment increasing or decreasing?
B) What is the initial value of the investment?
C) What is the effective annual rate of the account?

Question 28

A radioactive substance decays by 11% every year. Which of the following is the formula for the quantity, , of a 20 gram sample remaining after
years?

A)
B)
C)
D)

Question 29

The amount of pollution in a harbor t hours after it was contaminated by illegal dumping is given by tons. What percentage of the pollution leaves the harbor each hour?

Question 30

Write a formula that gives the value in an account after t years. Assume that the initial value in the account is $1200 and that the account doubles in value every 8 years.

Question 31

Write a formula that gives the value of an investment which is initially worth $125,000 and loses value at a rate of 2.9% per year.

Question 32

Kathleen opens a savings account with $1600. The account earns 3.6% annual interest compounded quarterly. How much will be in the account after 12 years?

Question 33

Kathleen opens a savings account with $1200. The account earns 3.3% annual interest compounded monthly. How much will be in the account after 16 years?

Question 34

Kathleen opens a savings account with $1200. The account earns 3.4% annual interest compounded weekly. How much will be in the account after 15 years?

Question 35

Suppose Taylor win $10,000 in a lottery. If she invests half in a CD earning 4.1% annual interest compounded quarterly and the rest in a savings account earning 3.9% annual interest compounded monthly. How much money does she have after 10 years?

Question 36

Taylor has $11,000 which she would like to invest. She can either invest in a CD earning 3.1% annual interest compounded quarterly, or she can deposit the money in a savings account earning 2.9% annual interest compounded monthly. If she chooses the investment with the maximum return, how much will she have at the end of 5 years?

Question 37

Taylor has $12,000 which she would like to invest. She can either invest in a CD earning 3.3% annual interest compounded quarterly, or she can deposit the money in a savings account earning 3.25% annual interest compounded monthly. If she chooses the investment with the maximum return, how much interest will she have earned at the end of 5 years?

Question 38

How much interest is earned in an account yielding 4.2% annual interest compounded weekly if the initial investment is $1000 and the money stays in the account for 20 years?

Question 39

Write a formula that gives the value of an investment which is initially worth $124,000 and gains value at a rate of 2.6% per year.

Question 40

An investment grows by 1.4% per year for 25 years. By what percent does it increase over the 25-year period? Give your answer correct to four decimal places.

Question 41

An investment decreases by 20% over a decade. By what effective annual rate does it decrease? Give your answer correct to four decimal places.

Question 42

Write a verbal description for each of the following formulas. The units are dollars and years.

Question 43

The price of an item increases due to inflation. Let Give the price of the item as a function of time in years, with t = 0 in 2004. What is the practical interpretation of ?

A) The price in 2004 of an item that is $75 now.
B) The price now of an item that was $75 in 2004.
C) The time at which the price reaches $75.
D) The price of the item after 75 years.

Question 44

The price of an item increases due to inflation. Let be the price of the item as a function of time in years, with t = 0 in 2004. What is the daily inflation rate? Round to 2 decimal places.

Question 45

One solution to the equation 4 + x = 4(2)^x is x = 0. Use your calculator to estimate the other solution to 2 decimal places.

Question 46

What is the maximum number of solutions the equation 4 + x = 4(2)^x can have?

Question 47

The graph of the exponential function is shown below. The formula for must be
.

Question 48

The graph of the exponential function Is shown below. Suppose Represents a city's population, in thousands, t years after 1995. Which of the following quantities are equivalent?

A)
B) Approximately -295 thousand
C) Approximately 705 thousand
D) The change in the city's population between 1996 and 2000
E)
F) The rate at which the population is declining between 1996 and 2000

Question 49

The US population in 2005 was approximately 296.4 million. Assume the population increases at a rate of 1.22% per year. Some demographers believe that the ideal population of the United States is about 130 million. According to this model, in what year did this occur?

Question 50

Question 51

Question 52

In the following figure, the functions f, g, h, and p can all be written in the form . Which one has the smallest value for b?

Question 53

Solve graphically for x if y = 13. Round to 1 decimal place.

Question 54

The price of an item increases due to inflation. Let give the price of the item as a function of time in years, with t = 0 in 2004. Estimate to 2 decimal places.

Question 55

Suppose the amount of ozone in the atmosphere is decreasing exponentially at a continuous rate of 0.26% per year. How many years will it take before one-fourth of the ozone has disappeared? Round to the nearest year.

Question 56

The population of a city is increasing exponentially. In 2000, the city had a population of 60,000. In 2005, the population was 83,748. Let be the population of the town t years after 2000. Use a graph of to estimate the year in which the population will reach 250,000.

Question 57

The following figure gives the graph of , where C is the computer hard disk capacity (in hundreds of megabytes) that could be bought for $500 t years past 1989. If the trend displayed in the graph continued, how many megabytes would a $500 hard drive have in 1999? Round to the nearest hundred.

Question 58

The following figure gives the graph of , where C is the computer hard disk capacity (in hundreds of megabytes) that could be bought for $500 t years past 1989. If the trend displayed in the graph continued, in what year would the capacity that can be bought for $500 be 9700?

Question 59

Which of the following characteristics describe the graph of ?

A) It is concave down.
B) It is concave up.
C) It crosses the y-axis at 1.
D)
E)
F)

G)

Question 60

The amount of pollution in a harbor t hours after it was contaminated by illegal dumping is given by tons. After how many hours is there less than 23 tons of pollution in the harbor? Round to 1 decimal place.

Question 61

What is the horizontal asymptote of

A) y = 0.85
B) y = 25
C) y = 0
D) There isn't one

Question 62

Consider the following figure, where Graph I has equation , Graph II has equation , Graph III has equation , and Graph IV has equation .
Is positive or negative?

Question 63

Let be the intersection of the graphs of the two exponential functions and , where . If b is decreased, does increase, decrease, or stay the same?

Question 64

Let be the intersection of the graphs of the two exponential functions and , where . If s is decreased, does increase, decrease, or stay the same?

Question 65

Assume that all important features are shown in the following graph of . What is ? For or , enter "inf" or "-inf".

Question 66

The graph of has a horizontal asymptote at . (If there is no horizontal asymptote, enter "DNE".)

Question 67

Which of the following formulas are exponential?

A)
B)
C)
D)
E)
F)
G)

Question 68

The number of books in a library tends to increase by the same percentage each year. Should a linear or an exponential function be used to model this scenario?

Question 69

The following table gives values from an exponential or a linear function. Determine which, and find values for a and b so that if the function is linear, or if the function is exponential.
a = ______, b = ______.

Question 70

The table below shows v, the dollar value of a share of a certain stock, as a function of t, the time (in weeks) since the initial offering of the stock. A possible formula for is . Round the second answer to 3 decimal places.

Question 71

A population has size 2300 at time t = 0, with t in years. If the population grows by 90 people per year, what is the formula for P, the population at time t?

A)
B)
C)
D)

Question 72

A population has size 4900 at time t = 0, with t in years. If the population grows by 15% per year, what is the formula for P, the population at time t?

A)
B)
C)
D)

Question 73

The graph below shows the quantity of a drug in a patient's bloodstream over a period of time t, in minutes.
Which of the following scenarios best describes the graph?

A) The drug is injected over a 10 minute interval, during which the quantity increases linearly. After the 10 minutes, the injection is discontinued and the quantity then decays exponentially.
B) The drug is injected over a 10 minute interval, during which the quantity increases exponentially. After the 10 minutes, the injection is discontinued and the quantity then decays linearly.
C) The drug is injected all at once. The quantity first increases and then decreases linearly.
D) The drug is injected all at once. The quantity first increases and then decreases exponentially.

Question 74

Kevin buys a new CD player for $190, and finds two years later when he wants to sell it that it is only worth $45. Assuming the value of the CD player decreases exponentially, the formula for , the value of the CD player after t years, is given by . Round your second answer to 2 decimal places.

Question 75

A biologist measures the amount of contaminant in a lake 3 hours after a chemical spill and again 11 hours after the spill. She sets up a possible model to determine Q, the amount of the chemical remaining in the lake as a function of t, the time in hours since the spill. The model assumes the contaminant is leaving the lake at a constant rate, which she determines to be 6 tons/hour. She estimates that the lake will be free from the contaminant 30 hours after the spill. How many tons of the contaminant were in the lake at the 3 hour reading?

Question 76

A biologist measures the amount of contaminant in a lake 5 hours after a chemical spill and again 8 hours after the spill. She sets up two possible models to determine Q, the amount of the chemical remaining in the lake as a function of t, the time in hours since the spill. The first model assumes the contaminant is leaving the lake at a constant rate, which she determines to be 3 tons/hour. Using this model, she estimates that the lake will be free from the contaminant 30 hours after the spill. Thus, and . The second model assumes that the amount of contaminant decreases exponentially. In this model, she finds that . Round both answers to 3 decimal places.

Question 77

A biologist measures the amount of contaminant in a lake 2 hours after a chemical spill and again 11 hours after the spill. She sets up two possible models to determine Q, the amount of the chemical remaining in the lake as a function of t, the time in hours since the spill. The first model assumes the contaminant is leaving the lake at a constant rate, which she determines to be 3 tons/hour. Using this model, she estimates that the lake will be free from the contaminant 25 hours after the spill. The second model assumes that the amount of contaminant decreases exponentially. She measures the spill a third time after 18 hours and finds that 29 tons remain. Which model seems best?

A) The linear one
B) The exponential one

Question 78

Question 79

Question 80

The following figure shows two functions, one linear and the other exponential. The formula for the linear one is , and the formula for the exponential one is . Round the first two answers to 2 decimal places and the last two answers to 3 decimal places.