Deck 4: Exponential Functions

Question 1

In the exponential formula $Q=47,200(0.9)^{t}$ , if $Q=a(1+r)^{t}$ then $r=\ldots \%$ .

Answer 1

-10

Question 2

In 2006, the cost of a particular piece of computer equipment was $\$ 490$ and going down at a rate of $14 \%$ per year. Assuming this percentage remains constant, what is the formula for $C$ , the cost of this equipment in dollars, as a function of $t$ , the number of years since 2006 ?

A) $C=490(0.86)^{t}$
B) $C=490(-1.14)^{t}$
C) $C=490(0.14)^{t}$
D) $C=490(-0.86)^{t}$

Answer 2

$C=490(0.86)^{t}$

Question 3

If you start with $\$ 8,000$ , how much money will you have after a $10 \%$ increase?

Answer 3

$\$ 8,800$

Question 4

If you start with $\$ 6,000$ , how much money will you have after a $5 \%$ increase followed by a $25 \%$ decrease?

Question 5

Which of the graphs in the following figure is the graph of $150(1.1)^{t}$ ?

Question 6

What is the growth factor if sales increase by $37 \%$ each month?

Question 7

The populations of 4 species of animals are given by the following equations:
$P_{1}=650(0.71)^{t} P_{2}=600(1.2)^{t} P_{3}=270(0.88)^{t} P_{4}=610(1.05)^{t}$
Which species are shrinking in size?

A) $P_{4}$
B) $P_{3}$
C) $P_{2}$
D) $P_{1}$

Question 8

The populations of 4 species of animals are given by the following equations:
$P_{1}=470(0.81)^{t} P_{2}=900(1.28)^{t} P_{3}=670(0.73)^{t} P_{4}=640(1.05)^{t}$
What is the annual percent growth rate for the population that is growing the fastest?

Question 9

The populations of 4 species of animals are given by the following equations:
$P_{1}=390(0.75)^{t} P_{2}=100(1.09)^{t} P_{3}=230(0.82)^{t} P_{4}=600(1.05)^{t}$
What is the largest initial population of the 4 species?

Question 10

The US population in 2005 was approximately 296.4 million. Assume the population increases at a rate of $1.38 \%$ per year. How many million people would you expect to live in the United States in the year 2020? Round to 1 decimal place.

Question 11

A quantity decreased from 100 to 95 . By what percentage did it decrease?

Question 12

A store's sales of cassette tapes of music decreased by $6 \%$ per year over a period of 5 years. By what total percent did sales of cassette tapes decrease over this time period? Round to 1 decimal place.

Question 13

The population of a city has been growing at a rate of $4 \%$ per year. If the population was 100,000 in 1990, what was the population in 1995? Round to the nearest whole number.

Question 14

In some developing countries, the inflation rate has been as high as $0.94 \%$ per day. What is the corresponding percent annual inflation rate? Round to the nearest whole number.

Question 15

The price of an item increases due to inflation. Let $p(t)=22.50(1.037)^{t}$ be the price of the item as a function of time in years, with $t=0$ in 2004. What is the weekly inflation rate? Round to 2 decimal places.

Question 16

The price of an item increases due to inflation. Let $p(t)=22.50(1.029)^{t}$ give the price of the item as a function of time in years, with $t=0$ in 2004. What is the practical interpretation of $p^{-1}(70)$ ?

A) The price in 2004 of an item that is $\$ 70$ now.
B) The price now of an item that was $70 in 2004.
C) The time at which the price reaches $\$ 70$ .
D) The price of the item after 70 years.

Question 17

The amount of pollution in a harbor $t$ hours after it was contaminated by illegal dumping is given by $A=70(0.65)^{t}$ tons. What percentage of the pollution leaves the harbor each hour?

Question 18

A radioactive substance decays by $12 \%$ every year. Which of the following is the formula for the quantity, $Q$ , of a 10 gram sample remaining after $t$ years?

A) $Q=10(0.88)^{t}$
B) $Q=10(1.12)^{t}$
C) $Q=10(-1.12)^{t}$
D) $Q=10(-0.88)^{t}$

Question 19

Which equation has a rate of change of $9 \%$ ?

A) $Q=(3,200) 0.91^{t}$
B) $Q=(3,200) 109^{t}$
C) $Q=(3,200) 1.09^{t}$
D) $Q=(3,200) 91^{t}$

Question 20

Match the graph to its equation.

A) $y=-1(0.8)^{x}$
B) $y=1(0.8)^{x}$
C) $y=1(1.2)^{-x}$
D) $y=1(1.2)^{x}$

Question 21

Match the graph to its equation.

A) $y=-4(1.1)^{x}$
B) $y=4(1.1)^{x}$
C) $y=4(0.9)^{-x}$
D) $y=4(0.9)^{x}$

Question 22

Match the graph to its equation.

A) $y=-4(1.1)^{x}$
B) $y=4(1.1)^{x}$
C) $y=4(0.9)^{-x}$
D) $y=4(0.9)^{x}$

Question 23

Which of the following formulas are exponential?

A) $P(t)=8.9(1.21)^{t}$
B) $P(t)=1.89-9.68 t$
C) $P(t)=17.8^{t}+40.05 t-22.25$
D) $P(t)=(1.4641)^{t} 2670$
E) $P(t)=178 e^{2.42 t}$
F) $P(t)=1.78 e^{-0.605 t}$
G) $P(t)=(4.45+12.1 t)^{t}$

Question 24

The average life expectancy in a country tends to increase by the same percentage each year. Should a linear or an exponential function be used to model this scenario?

Question 25

The following table gives values from an exponential or a linear function. Determine which, and find values for $a$ and $b$ so that $f(x)=a+b x$ if the function is linear, or $f(x)=a(b)^{x}$ if the function is exponential.
a= ---------------,b= ------------

Question 26

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 $v(t)$ is $v(t)=\ldots(\ldots)^{t}$ . Round the second answer to 3 decimal places.

Question 27

A population has size 3,500 at time $t=0$ , with $t$ in years. If the population grows by 80 people per year, what is the formula for $P$ , the population at time $t$ ?

A) $P=3,500(1.8) t$
B) $P=3,500+80 t$
C) $P=3,500(0.8)^{t}$
D) $P=3,500(1.8)^{t}$

Question 28

A population has size 3,000 at time $t=0$ , with $t$ in years. If the population grows by $10 \%$ per year, what is the formula for $P$ , the population at time $t$ ?

A) $P=3,000(1.1) t$
B) $P=3,000+10 t$
C) $P=3,000(1.1)^{t}$
D) $P=3,000(0.1)^{t}$

Question 29

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 30

Kevin buys a new CD player for $\$ 300$ , and finds two years later when he wants to sell it that it is only worth $\$ 82$ . Assuming the value of the CD player decreases exponentially, the formula for $V(t)$ , the value of the CD player after $t$ years, is given by $V(t)=\ldots(\ldots)^{t}$ . Round your second answer to 2 decimal places.

Question 31

A biologist measures the amount of contaminant in a lake 2 hours after a chemical spill and again 15 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 35 hours after the spill. How many tons of the contaminant were in the lake at the 15 hour reading?

Question 32

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 20 hours after the spill. Thus, Q(2)= ---------- and Q(11)= ----------- The second model assumes that the amount of contaminant decreases exponentially. In this model, she finds that $Q(t)=\ldots(\ldots)^{t}$ . Round both answers to 3 decimal places.

Question 33

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 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 7 tons\hour. Using this model, she estimates that the lake will be free from the contaminant 24 hours after the spill. The second model assumes that the amount of contaminant decreases exponentially. She measures the spill a third time after 23 hours and finds that 44 tons remain. Which model seems best?

A) The linear one
B) The exponential one

Question 34

Let $f(x)$ be given in the table below. Find the value of $k$ if $f(x)$ is linear.

Question 35

Let $f(x)$ be given in the table below. Find the value of $k$ if $f(x)$ is exponential.

Question 36

The following figure shows two functions, one linear and the other exponential. The formula for the linear one is f(x) =-----------+------------x, and the formula for the exponential one is $g(x)=\ldots(\ldots)^{x}$ . Round the first two answers to 2 decimal places and the last two answers to 3 decimal places.

Question 37

Each of the functions in the table below is increasing, but each increases in a different way. One is linear, one is exponential, and one is neither.

Which one is exponential: $f, g$ , or $h$ ?

Question 38

Each of the functions in the table below is increasing, but each increases in a different way. One is linear, one is exponential, and one is neither.

The following three graphs correspond with the functions in the table. Which is the graph of $g$ ?

A)

B)

C)

Question 39

Each of the functions in the table below is increasing, but each increases in a different way. One is linear, one is exponential, and one is neither.

The formula for the exponential one is $\ldots(\ldots)^{t}$ Round your second answer to 2 decimal places.

Question 40

The population of a city is increasing exponentially. In 2000 , the city had a population of 40,000 . In 2005 , the population was 58,502. The formula for $P(t)$ , the population of the town $t$ years after 2000 , is given by $p(t)=\ldots(\ldots)^{t}$ .Round your second answer to 3 decimal places.

Question 41

The formula for the exponential function $P$ such that $P(11)=10$ and $P(12)=9$ is given by $P(t)=\ldots(\ldots)^{t}$ . Give both answers to 3 decimal places.

Question 42

Find a possible formula for the exponential function $f$ such that the points $(5,2684.355)$ and $(3,262.144)$ are on the graph.

Question 43

Write the formula for the price $p$ of a gallon of gas in $t$ days if the price is $\$ 3.85$ on day $t$ $=0$ and the price increases by $\$ 0.08$ per day.

Question 44

Write the formula for the price $p$ of a gallon of gas in $t$ days if the price is $\$ 3.35$ on day $t$ $=0$ and the price increases by $7 \%$ per day.

Question 45

One solution to the equation $3+x=3(2)^{x}$ 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 $2+x=2(4)^{x}$ can have?

Question 47

The graph of the exponential function $P(t)$ is shown below. The formula for $p(t)=\ldots(\ldots)^{t}$

Question 48

The graph of the exponential function $P(t)$ is shown below. Suppose $P(t)$ represents a city's population, in thousands, $t$ years after 1995. Which of the following quantities are equivalent?

A) $P(7)-P(4)$
B) Approximately -125 thousand
C) Approximately 387 thousand
D) The change in the city's population between 1999 and 2002
E) $P(7)/P(4)$
F) The rate at which the population is declining between 1999 and 2002

Question 49

The US population in 2005 was approximately 296.4 million. Assume the population increases at a rate of $1.34 \%$ 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

Find $\lim _{t \rightarrow \infty} 20,000(0.82)^{t}$ . For $\infty$ or $-\infty$ , enter "inf" or "-inf".

Question 51

Find $\lim _{x \rightarrow-\infty} 7.2 e^{0.11 x}$ . For $\infty$ or $-\infty$ , enter "inf" or "-inf".

Question 52

In the following figure, the functions $f, g, h$ , and $p$ can all be written in the form $y=a b^{t}$ . Which one has the largest value for $b$ ?

Question 53

Solve $y=18(0.84)^{x}$ graphically for $x$ if $y=13$ . Round to 2 decimal places.

Question 54

The price of an item increases due to inflation. Let $p(t)=12.50(1.024)^{t}$ give the price of the item as a function of time in years, with $t=0$ in 2004 . Estimate $p^{-1}(75)$ to 2 decimal places.

Question 55

Suppose the amount of ozone in the atmosphere is decreasing exponentially at a continuous rate of $0.27 \%$ per year. How many years will it take before one-third 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 70,000. In 2003, the population was 89,870 . Let $P(t)$ be the population of the town $t$ years after 2000 . Use a graph of $P(t)$ to estimate the year in which the population will reach 250,000 .

Question 57

The following figure gives the graph of $C=f(t)$ , where $\mathrm{C}$ 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 1997? Round to the nearest hundred.
$C$ , capacity (in 100 s of megabytes)

Question 58

The following figure gives the graph of $C=f(t)$ , where $\mathrm{C}$ 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 4,600 ?
$C \text {, capacity (in } 100 \text { s of megabytes) }$

Question 59

Which of the following characteristics describe the graph of $f(x)=(1.5)^{x}$ ?

A) It is concave down.
B) It is concave up.
C) It crosses the $y$ -axis at 1 .
D) As $x \rightarrow \infty, f(x) \rightarrow \infty$ .
E) As $x \rightarrow \infty, f(x) \rightarrow 0$ .
F) As $x \rightarrow-\infty, f(x) \rightarrow \infty$ .
G) As $x \rightarrow-\infty, f(x) \rightarrow 0$ .

Question 60

The amount of pollution in a harbor $t$ hours after it was contaminated by illegal dumping is given by $A=50(0.8)^{t}$ tons. After how many hours is there less than 10 tons of pollution in the harbor? Round to 1 decimal place.

Question 61

What is the horizontal asymptote of $y=87(0.64)^{t}$ as $t \rightarrow \infty$ ?

A) $y=0.64$
B) $y=87$
C) $y=0$
D) There isn't one

Question 62

Consider the following figure, where Graph I has equation $y=a_{1} e^{b_{1} x}$ , Graph II has equation $y=a_{2} e^{b_{2} x}$ , Graph III has equation $y=a_{3} e^{b_{3} x}$ , and Graph IV has equation $y=a_{4} e^{b_{4} x}$ .

Is $b_{1}$ positive or negative?

Question 63

Let $\left(x_{0}, y_{0}\right)$ be the intersection of the graphs of the two exponential functions $y=a e^{b x}$ and $y=c e^{d x}$ , where $0<a<c$ . If $a$ is increased, does $x_{0}$ increase, decrease, or stay the same?

Question 64

Let $\left(t_{0}, P\left(t_{0}\right)\right)$ be the intersection of the graphs of the two exponential functions $P=a(1+r)^{t}$ and $P=b(1+s)^{t}$ , where $0<a<b$ . If $r$ is increased, does $P\left(t_{0}\right)$ increase, decrease, or stay the same?

Question 65

Assume that all important features are shown in the following graph of $y=f(x)$ .
What is $\lim _{x \rightarrow-\infty} f(x)$ ? For $\infty$ or $-\infty$ , enter "inf" or "-inf".

Question 66

The graph of $P(t)=1.7^{-t}+29$ has a horizontal asymptote at $P(t)=$ ---------. (If there is no horizontal asymptote, enter "DNE".)

Question 67

Is the function graphed exponential?

Question 68

Is the function graphed exponential?

Question 69

Solve $y=5(1.1)^{x}$ for $x$ if $y=6.655$ .

Question 70

An investment grows according to the formula $V=7000 e^{0.051 t}$ . How many years will it take for the original investment to triple? Round to 1 decimal place.

Question 71

The price of an item increases due to inflation. Let $p(t)=32.50(1.047)^{t}$ 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.

Question 72

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
$f(t)=r(1+N)^{t} ?$

Question 73

Is the formula for a function representing a quantity which begins at $4 N$ in year $t=0$ and grows at a continuous annual rate of $r \%$ given by
$f(t)=4 N e^{r t /100} ?$

Question 74

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 $f(t)=1.35 N e^{r t /100}$ ?

Question 75

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 $\frac{r}{3} \%$ given by
$f(t)=N e^{\frac{r t}{3}} ?$

Question 76

Let $P(t)=4,500 e^{0.043 t}$ give the size of a population of animals in year $t$ . What will the population be after 12 years? Round to the nearest whole number.

Question 77

Let $P(t)=5,000 e^{0.037 t}$ give the size of a population of animals in year $t$ . After how many years will the population be approximately 10,099 ? Round to the nearest year.

Question 78

The US population in 2005 was approximately 296.4 million. Assume the population increases at a rate of $1.34 \%$ per year. What is the formula for $P(t)$ , the population for the United States $t$ years after 2005?

A) $P(t)=296.4(0.0134)^{t}$
B) $P(t)=296.4(1.0134)^{t}$
C) $P(t)=296.4 e^{1.0134 t}$
D) $P(t)=296.4 e^{0.0134 t}$

Question 79

The figure below shows the graphs of the following functions:
(A) $y=e^{t}$
(B) $y=2^{t}$
(C) $y=e^{-t}$
(D) $y=2^{-t}$

Which one is the graph of A?

Question 80

What is $\lim _{x \rightarrow \infty} e^{-4 x}$ ? If necessary, enter "inf" for $\infty$ and "-inf" for $-\infty$ .