Resources

Resources

Write exponential functions: word problems

A biology class at Central High School predicted that a local population of animals will double in size every

12

years. The population at the beginning of

2014

was estimated to be

50

P

represents the population

n

2014

, then which of the following equations represents the class’s model of the population over time?

\newline

P = 12 + 50n

\newline

P = 50 + 12n

\newline

P = 50(2)^{12n}

\newline

P = 50(2)^{\frac{n}{12}}

The number of users on a website is

6300

and is growing exponentially at a rate of

73 \%

per year. Write a function to represent the number of users on the website after

t

years, where the daily rate of change can be found from a constant in the function. Round all coefficients in the function to four decimal places. Also, determine the percentage rate of change per day, to the nearest hundredth of a percent.

Increases in snow levels are recorded with positive numbers. Decreases in snow levels are recorded with negative numbers.

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After a winter storm, the snow on Cherry Street started melting at a rate of

\frac{1}{3} \mathrm{~cm}

\newline

What was the total change in depth of the snow on Cherry Street after

3

\newline

Increases in temperature are recorded with positive numbers. Decreases in temperature are recorded with negative numbers.

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Once the sun went down, the outside temperature decreased

2^{\circ} \mathrm{C}

4

\newline

What is the total change in temperature over the

4

\newline

{ }^{\circ} \mathrm{C}

The histogram below shows information about the daily energy output of a solar panel for a number ot days.

\newline

Calculate an estimate for the mean daily energy output.

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If your answer is a decimal, give it to

1

Susan bought a new car in

2020

\$48000

. If the value of the car decreases by

11\%

each year write an exponential modle for the value of the car. Then, estimate the cost of the car in the year

2023

21

A conservation agency tracks the sea turtle population by counting the number of nesting sites where the turtles lay their eggs. The table shows the numbers of nesting sites for several years since

2001

. The data can be modeled by an exponential function.

\newline

\newline

\begin{tabular}{|c|c|}

\newline

\hline \begin{tabular}{c}

\newline

Number of Years \\

\newline

2001, x

\newline

\end{tabular} & \begin{tabular}{c}

\newline

Number of Nesting \\

\newline

n(x)

\newline

\end{tabular} \\

\newline

1

46

125

\newline

2

37

994

\newline

3

40

513

\newline

4

29

368

\newline

5

34

082

\newline

6

31

746

\newline

7

27

691

\newline

\newline

\newline

Which function best models the data?

\newline

n(x)=25,956.80(1.08)^{x}

\newline

n(x)=46,797.94(0.93)^{x}

\newline

n(x)=1.08(25,956.80)^{x}

This exercise uses the radioactive decay model.

\newline

The half-life of strontium

-90

29

years. How long (in yr) will it take a

70

-milligram sample to decay to a mass of

56

mg? (Round your answer to the nearest whole number.)

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The population of a city decreases by

1 \%

per year. What should we multiply the current population by to find next year's population in one step?

\newline

The population of a city increases by

4 \%

per year. If this year's population is

p

, which expression represents next year's population?

\newline

1.004 p

\newline

1.04 p

\newline

4 p

\newline

0.04 p

The population of a city decreases by

1 \%

per year. If this year's population is

p

, which expression represents next year's population?

\newline

0.99 p

\newline

0.999 p

\newline

0.01 p

\newline

1 p

The population of a city increases by

4 \%

per year. If this year's population is

p

, which expression represents next year's population?

\newline

p+0.04 p

\newline

0.04 p

\newline

1+0.04 p

\newline

4 p

Write an exponential function to model the situation. The value of a car is

\$ 18000

and is depreciating at a rate of

12 \%

\newline

V(t)=18000(.88)^{t}

\newline

V(t)=.88(18000)^{t}

\newline

V(t)=.12(18000)^{t}

\newline

V(t)=18000(.12)^{t}

The table below shows the amount of lemon juice and sugar needed to make three different-sized batches of lemonade using the same recipe.

\newline

\begin{tabular}{lcc|}

\newline

& Lemon juice (mL) & Sugar (g) \\

\newline

\hline Batch A &

500

200

\newline

750

300

\newline

1500

600

\newline

\newline

\newline

Write an equation to describe the relationship between

j

, the amount of lemon juice in milliliters, and

s

, the amount of sugar in grams.

A lake near the Arctic Circle is covered by a thick sheet of ice during the cold winter months. When spring arrives, the warm air gradually melts the ice, causing its thickness to decrease at a rate of

0.2

meters per week. After

7

weeks, the sheet is only

2.4

\newline

y

represent the ice sheet's thickness (in meters) after

x

\newline

Complete the equation for the relationship between the thickness and number of weeks.

\newline

y=\square

Eagle Eye Tour Company plans to offer scenic helicopter tours as its newest attraction. The company spent

\$979,000

on a pre-owned helicopter, which is expected to lose about

11\%

of its value each year. Write an exponential equation in the form

y=a(b)^x

that can model the value of the helicopter,

y

x

years after purchase. Use whole numbers, decimals, or simplified fractions for the values of

a

b

y =

To the nearest hundred dollars, how much will the helicopter be worth

6

years after purchase?

170

deer on a reservation. The deer population is increasing at a rate of

30\%

per year. Write a function that gives the deer population

p(t)

on the reservation

t

years from now.

=1

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2

\newline

3

\newline

4

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5

\newline

6

\newline

The next model of a sports car will cost

12.6 \%

less than the current model. The current model costs

\$ 58,000

. How much will the price decrea will be the price of the next model?

\newline

Decrease in price:

\quad \$ \square

\newline

Price of next model:

\newline

Chloe has a collection of vintage action figures that is worth

\$ 310

. If the collection appreciates at a rate of

15 \%

per year, which equation represents the value of the collection after

5

\newline

V=310(1.15)^{5}

\newline

V=310(1+0.15)

\newline

V=310(1-0.15)^{5}

\newline

V=310(0.85)^{5}

In a lab experiment, a population of

250

bacteria is able to triple every hour. Which equation matches the number of bacteria in the population after

2

\newline

B=250(3)^{2}

\newline

B=3(250)^{2}

\newline

B=3(250)(250)

\newline

B=250(3)(3)(3)(3)

A radioactive compound with mass

280

grams decays at a rate of

18.3 \%

per hour. Which equation represents how many grams of the compound will remain after

2

\newline

C=280(1.183)^{2}

\newline

C=280(1-0.183)(1-0.183)

\newline

C=280(1+0.183)^{2}

\newline

C=280(1-0.183)(1-0.183)(1-0.183)(1-0.183)

A town has a population of

36

500

and grows at a rate of

6.4 \%

every year. Which equation represents the town's population after

5

\newline

P=36,500(1+0.064)

\newline

P=36,500(1-0.064)^{5}

\newline

P=36,500(0.936)^{5}

\newline

P=36,500(1+0.064)^{5}

A radioactive compound with mass

330

grams decays at a rate of

21 \%

per hour. Which equation represents how many grams of the compound will remain after

4

\newline

C=330(1-0.21)

\newline

C=330(1.21)^{4}

\newline

C=330(1+0.21)^{4}

\newline

C=330(1-0.21)(1-0.21)(1-0.21)(1-0.21)

A radioactive compound with mass

420

grams decays at a rate of

4 \%

per hour. Which equation represents how many grams of the compound will remain after

5

\newline

C=420(0.6)^{5}

\newline

C=420(1+0.04)^{5}

\newline

C=420(1-0.04)(1-0.04)(1-0.04)

\newline

C=420(0.96)^{5}

A group consisting of

8

aggressive zombies triples in size every hour. Which equation matches the number of zombies after

5

\newline

Z=3(1+8)^{5}

\newline

Z=3(8)^{5}

\newline

Z=8(3)^{5}

\newline

Z=8(3)

Kamron's house is

0.5

mile from the grocery store. If Kamron's house is

1.5

miles closer to the grocery store than Amy's house is, which of the following can be used to determine

x

, the distance between Amy's house and the grocery store in miles?

\newline

1

\newline

0.5 \times x=1.5

\newline

1.5-x=0.5

\newline

x-1.5=0.5

\newline

x+0.5=1.5

Complet the T chart of values for the Exponential equation. Round to the nearest

10^{th}

\newline

When you have checked your quiz and know that your T-Chart is correct, graph this Exponential equation and upload it (along with the others) to the Absolute Value and Exponential Graph Upload Assignment for points

\newline

y=3^{x}+4

\newline

\newline

x

\newline

y

\newline

\newline

0

\newline

\newline

1

\newline

=

\newline

2

\newline

=

\newline

3

\newline

=

\newline

y=3^{x}+4

1

\newline

\newline

y=3^{x}+4

2

Complet the T chart of values for the Exponential equation. Round to the nearest

10^{th}

\newline

When you have checked your quiz and know that your T-Chart is correct, graph this Exponential equation and upload it (along with the others) to the Absolute Value and Exponential Graph Upload Assignment for points

\newline

y=3^{z}+4

\newline

T chart of values for the Exponential equation. Round to the nearest

10^{th}

\newline

When you have checked your quiz and know that your T-Chart is correct, graph this Exponential equation and upload it (along with the others) to the Absolute Value and Exponential Graph Upload Assignment for points

\newline

y=-3^{x}

\newline

\newline

x

\newline

y

\newline

\newline

0

\newline

\newline

1

\newline

\newline

2

\newline

\newline

3

\newline

\newline

-1

\newline

\newline

-2

Complet the T chart of values for the Exponential equation. Round to the nearest

10

th if necessary.

\newline

When you have checked your quiz and know that your T-Chart is correct, graph this Exponential equation and upload it (along with the others) to the Absolute Value and Exponential Graph Upload Assignment for points

\newline

y=3^{x}

\newline

\begin{array}{c|c}

\newline

\newline

0

1

2

3

-1

-2

& (\newline\)\end{array}

3

Write exponential functions: word problems DBW.

\newline

Lakewood Children's Theater started a summer program last year with

245

students. Thanks to some advertising during the off-season,

294

students have enrolled this year. The theater will continue this advertising strategy in hopes that enrollment will continue to increase.

\newline

Write an exponential equation in the form

y=a(b)^{x}

that can model the number of enrolled students,

y, x

years after the program began.

\newline

Use whole numbers, decimals, or simplified fractions for the values of

a

b

\newline

y=

1950

, the per capita gross domestic product (GDP) of Australia was approximately

\$1800

. Each year afterwards, the per capita GDP increased by approximately

6.7\%

. Write a function that gives the approximate per capita GDP

G(t)

t

Kamron's house is

0.5

mile from the grocery store. If Kamron's house is

1.5

miles closer to the grocery store than Amy's house is, which of the following can be used to determine

x

, the distance between Amy's house and the grocery store in miles?

\newline

1

\newline

0.5 \times x=1.5

\newline

1.5-x=0.5

\newline

x-1.5=0.5

\newline

x+0.5=1.5

The rate of change

\frac{d P}{d t}

of the number of fox at a national park is modeled by a logistic differential equation. The maximum capacity of the park is

896

8

PM, the number of fox at the national park is

184

and is increasing at a rate of

28

fox per day. Write a differential equation to describe the situation.

\newline

\frac{d P}{d t}=\square

The rate of change

\frac{d P}{d t}

of the number of people infected by a disease is modeled by a logistic differential equation. The maximum capacity of the village is

943

8

PM, the number of people infected by the disease is

106

and is increasing at a rate of

30

people per hour. Write a differential equation to describe the situation.

\newline

\frac{d P}{d t}=\square

The rate of change

\frac{d P}{d t}

of the number of deer on an island is modeled by a logistic differential equation. The maximum capacity of the island is

541

12 \mathrm{PM}

, the number of deer on the island is

228

and is increasing at a rate of

15

deer per day. Write a differential equation to describe the situation.

\newline

\frac{d P}{d t}=\square

The rate of change

\frac{d P}{d t}

of the number of deer on an island is modeled by a logistic differential equation. The maximum capacity of the island is

673

8 \mathrm{PM}

, the number of deer on the island is

172

and is increasing at a rate of

37

deer per day. Write a differential equation to describe the situation.

\newline

\frac{d P}{d t}=\square

The rate of change

\frac{d P}{d t}

of the number of wolves at a national park is modeled by a logistic differential equation. The maximum capacity of the park is

955

8

PM, the number of wolves at the national park is

218

and is increasing at a rate of

26

wolves per day. Write a differential equation to describe the situation.

\newline

\frac{d P}{d t}=\square

The rate of change

\frac{d P}{d t}

of the number of students who heard a rumor is modeled by a logistic differential equation. The maximum capacity of the school is

942

5 \mathrm{AM}

, the number of students who heard the rumor is

233

and is increasing at a rate of

37

students per hour. Write a differential equation to describe the situation.

\newline

\frac{d P}{d t}=\square

The rate of change

\frac{d P}{d t}

of the number of deer on an island is modeled by a logistic differential equation. The maximum capacity of the island is

593

5 \mathrm{PM}

, the number of deer on the island is

210

and is increasing at a rate of

16

deer per day. Write a differential equation to describe the situation.

\newline

\frac{d P}{d t}=\square

The rate of change

\frac{d P}{d t}

of the number of people on an island is modeled by a logistic differential equation. The maximum capacity of the island is

745

11

AM, the number of people on the island is

186

and is increasing at a rate of

24

people per hour. Write a differential equation to describe the situation.

\newline

\frac{d P}{d t}=\square

Wangari plants trees at a constant rate of

12

3

\newline

Write an equation that relates

p

, the number of trees Wangari plants, and

h

, the time she spends planting them in hours.

Mia tried to solve an equation step by step.

\newline

0.1 t-3=5

\newline

0.1 t=2 \quad

1

\newline

t=20 \quad \text { Step } 2

\newline

Find Mia's mistake.

\newline

1

\newline

1

\newline

2

\newline

(C) Mia did not make a mistake.

Kwame must earn more than

16

stars per day to get a prize from the classroom treasure box.

\newline

Write an inequality that describes

S

, the number of stars Kwame must earn per day to get a prize from the classroom treasure box.

4.5 \mathrm{~km}

with the current in the same amount of time as it took her to paddle

3 \mathrm{~km}

against the current. Naira paddled at an average rate of

5 \frac{\mathrm{km}}{\mathrm{h}}

relative to the water each way. Assume the speed of the current was constant.

\newline

What was the speed of the current?

\newline

\square \frac{\mathrm{km}}{\mathrm{h}}

After a special medicine is introduced into a petri dish containing a bacterial culture, the number of bacteria remaining in the dish decreases rapidly.

\newline

The population loses

\frac{1}{4}

of its size every

44

seconds. The number of remaining bacteria can be modeled by a function,

N

, which depends on the amount of time,

t

\newline

Before the medicine was introduced, there were

11

880

bacteria in the Petri dish.

\newline

Write a function that models the number of remaining bacteria

t

seconds since the medicine was introduced.

\newline

N(t)=\square

Nicholas sent a chain letter to his friends, asking them to forward the letter to more friends. Every

12

weeks, the number of people who receive the email increases by an additional

99 \%

, and can be modeled by a function,

P

, which depends on the amount of time,

t

\newline

Nicholas initially sent the chain letter to

50

\newline

Write a function that models the number of people who receive the email

t

weeks since Nicholas initially sent the chain letter.

\newline

P(t)=\square

On the first day of spring, an entire field of flowering trees blossoms. The population of locusts consuming these flowers rapidly increases as the trees blossom. The locust population gains

87 \%

of its size every

2

4

days, and can be modeled by a function,

L

, which depends on the amount of time,

t

\newline

Before the first day of spring, there were

1100

locusts in the population.

\newline

Write a function that models the locust population

t

days since the first day of spring.

\newline

L(t)=

Bela started studying how the number of branches on her tree grows over time. Every

2

9

years, the number of branches increases by an additional

83 \%

, and can be modeled by a function,

N

, which depends on the amount of time,

t

\newline

When Bela began the study, her tree had

60

\newline

Write a function that models the number of branches

t

years since Bela began studying her tree.

\newline

N(t)=\square

-14

is an element which loses

\frac{1}{10}

of its mass every

871

years. The mass of a sample of carbon

-14

can be modeled by a function,

M

, which depends on its age,

t

\newline

We measure that the initial mass of a sample of carbon

-14

960

\newline

Write a function that models the mass of the carbon-

14

sample remaining

t

years since the initial measurement.

\newline

M(t)=