Additional PracticeGrowing,Growing,Growing

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Investigation 1

1. Cut a sheet of paper into fourths. Stack the four pieces and cut the stack into fourths. Stack all the pieces and cut that stack into fourths again.

How many pieces of paper would you have at the end of

a. Step 1? b. Step 2? c. Step 3?

d. Step 10? e. Step n?

For Exercises 2–5, write the expression in standard form.

2. 2¹⫻ 5¹ 3. 2²⫻ 5²

4. 2³⫻ 5³ 5. 2⁴⫻ 5⁴

Step 1 Step 2 Step 3

Name ____________________________________________ Date ____________ Class ____________

Additional Practice

Growing, Growing, Growing

000200010271960391_Unit3_Inv1-5_p080-106.qxd 11/25/15 1:45 AM Page 80

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81

Name ____________________________________________ Date ____________ Class ____________

Additional Practice (continued)

Growing, Growing, Growing Investigation 1

6. Suppose you drew a pattern of branching lines starting with 3 lines. Using a second color, you added 3 branches to the end of each of the first 3 lines:

Using a third color, you added 3 branches to the end of each of the 9 new lines.

a. Complete the table to show the number of branches you would draw in each new color.

b. Write an equation showing the relationship between the number of branches drawn b and the number of the color c.

c. What is the number of the first color with which you will draw at least 1,000 branches?

d. Make a graph of the (color, branches) data from part (a).

Color 1 2 3 4 5 6

Branches 3 9

000200010271960391_Unit3_Inv1-5_p080-106.qxd 11/24/15 7:50 PM Page 81

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Investigation 1 Name ____________________________________________ Date ____________ Class ____________

Additional Practice: Digital Assessments

9. A phone tree is planned to get the word out as quickly as possible about a schedule change at school. Suppose Step 1 is the lead person calling 5 people. Step 2 is those 5 people each calling 5 new people. This process continues until all the people on the phone tree have been called.

Circle the number or expression that makes each statement true.

After Step 1,

Q

¹⁵¹⁰

U

new person/people know about the schedule change.

After Step 2,

Q

⁵¹⁰²⁵⁵⁰

U

new people know about the schedule change.

After Step 3,

Q

²⁵⁵⁰¹²⁵

U

After Step n,

Q

²^{5 3 n}⁵²ⁿ

U

7. Which of the following are equivalent to 3²3 2⁴? Select all that apply.

n 48 n 432 n 3 3 3 3 4 ² n 3 3 2 3 2 3 4

n 3 3 3 3 2 3 2 3 2 3 2

8. Circle the expression or number that makes each statement true.

The expression 60 ² can be written in standard form as Q^{60 3 2}^{60 3 60}U^.

The expression 60 ² is equivalent to

Q

¹²⁰^3,600

U

^andQ1.2 3 10 ² 3.6 3 10 ³ U^.

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83

Investigation 1

Write each expression in exponential form.

1. 3 ⫻ 3 ⫻ 3 ⫻ 3 ⫻ 3 2. 2.7 ⫻ 2.7 ⫻ 2.7

3. 2 ⫻ 2 ⫻ 2 ⫻ 2 ⫻ 2 ⫻ 2 4. 4 ⫻ 4 ⫻ 4 ⫻ 4 ⫻ 4 ⫻ 4 ⫻ 4 ⫻ 4

Write each expression in standard form.

5. (0.5)³ 6. (2.7)²

7. 2³ 8. (8.1)³

Write each number in scientific notation.

9. 480,000 10. 960,000

11. 8,750,000 12. 407,000

Name ____________________________________________ Date ____________ Class ____________

Skill: Using Exponents

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Growing, Growing, Growing 1. A bathtub is being filled at a rate of 2.5 gallons per minute. The bathtub will

hold 20 gallons of water.

a. How long will it take to fill the bathtub?

b. Is the relationship described linear, inverse, exponential, or neither? Write an equation relating the variables.

2. Suppose a single bacterium lands on one of your teeth and starts reproducing by a factor of 4 every hour.

a. After how many hours will there be at least 1,000,000 bacteria in the new colony?

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85

Investigation 2

3. Two students who work in a grocery store are making a display of canned goods. They build a tower of cans 12 layers deep. The first layer, at the top, contains three cans in a row. The second layer contains six cans, in two rows of three that support the first layer. The third layer has nine cans, in three rows of three that support the second layer.

a. How many cans are in layer 12, the bottom layer?

4. An experimental plant has an unusual growth pattern. On each day, the plant doubles its height of the previous day. On the first day of the experiment, the plant grows to twice, or 2 times, its original height. On the second day, the plant grows to 4 times its original height. On the third day, the plant grows to 8 times its original height.

a. How many times its original height does the plant reach on the sixth day?

On the nth day?

b. If the plant is 128 centimeters tall on the ninth day, how tall was it just before the experiment began?

c. Is the relationship described linear, inverse, exponential, or neither? Write an equation relating the variables.

layer 1

layer 2

Name ____________________________________________ Date ____________ Class ____________

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Growing, Growing, Growing Study the pattern in each table. Tell whether the relationship between x and y is

linear, inverse, exponential, or neither, and explain your reasoning. If the relationship is linear, inverse, or exponential, write an equation for it.

5.

6.

7.

8.

9. x y

0 1

1 14

2 116

3 614

4 2,156

5 10,124 x

y 0 1

1 2 3 4 5

1

2 1

3 1

4 1

5 1

6

x y

0 1 2

1 3 4

4 16

5

1 64

4 1 16

x y

0 2

1 4

2 8

3 16

4 32

5 64 x

y 0 2

1 9

2 16

3 23

4 30

5 37

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87

Name ____________________________________________ Date ____________ Class ____________

10. Which of the following statements describe the graph of y 5 3 · 4 ^x ? Select all that apply.

n The graph represents exponential growth.

n The y-intercept is (^{0, 1})^.

n There is no x-intercept.

n The graph passes through the point (1, 12).

11. Suppose an investment increases in value 2.5 times every 8 years. Complete the table using the tiles.

$1,000

$1,250

$3,125

$2,000

$7,812.50

Time (years)

Initial 8 16 24

$500 Value of Investment

12. An empty tank that holds 18,000 gallons of water is being filled from a pipe at a rate of 30 gallons per minute. Circle the number, word, or expression that makes each statement true.

There will be

Q

³⁰⁰⁶⁰⁰^1,800

U

gallons of water in the tank after 10 minutes.

The tank will be filled after

Q

³⁰³⁰⁰⁶⁰⁰

U

^minutes.

This relationship can be described as

Q

^linearexponential inverse

U

^.

An equation to represent the number of gallons of water, g, in the tank after

t minutes is g 5

Q

^30t³⁰^{30 1 t}^t

U

^.

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Investigation 2

Complete the table for each exercise.

1. An investment increases by 1.5 times every 5 years.

2. The number of animals doubles every 3 months.

3. The amount of matter doubles every 6 months.

Time Amount of (years) Matter (grams)

Initial 10

1 40

2 160

3

Time Number of (months) Animals

Initial 18

3 36

6 72

9 12

Time Value of (years) Investment

Initial $800

5 $1,200

10 $1,800

15 $2,700

20 25

Name ____________________________________________ Date ____________ Class ____________

Skill: Exponential Functions

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89

Investigation 2

Graph each function.

4. y 3^x

5. y 10 5^x

6. y 2¹₈ ^x

Name ____________________________________________ Date ____________ Class ____________

Skill: Exponential Functions (continued)

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Investigation 2

1. Suppose you deposit $1,000 in a savings account that earns interest of 6% per year on the current balance in the account.

a. If you leave your money in the account for 10 years, what will the value of your investment be at the end of the 10 years?

b. Write an equation relating the variables.

2. Janelle deposits $2,000 in the bank. The bank will pay 5% interest per year, compounded annually. This means that Janelle’s money will grow by 5% each year.

a. Make a table showing Janelle’s balance at the end of each year for 5 years.

b. Write an equation for calculating the balance b at the end of any year t.

c. Approximately how many years will it take for the original deposit to double in value? Explain your reasoning.

d. Suppose the interest rate is 10%. Approximately how many years will it take for the original deposit to double in value? How does this interest rate compare with an interest rate of 5%?

Name ____________________________________________ Date ____________ Class ____________

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91

Investigation 2

For Exercises 3–6, tell whether the relationship between x and y is linear, inverse, exponential, or neither, and explain your answer. If the relationship is linear, inverse, or exponential, write an equation for the relationship.

3.

4.

5.

6. _x y

1 2 3 4 5

1

2 1

4 1

6 1

8 1

10

x y

0 2.3

1 3.8

2 5.3

3 6.8

4 8.3

5 9.8 x

y 0 500

1 550

2 605

3 665.5

4 732.05

5 805.255 x

y 0 2

1 2.6

2 3.38

3 4.394

4 5.7122

5 7.42586

Name ____________________________________________ Date ____________ Class ____________

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Investigation 3

7. Consider these three equations: y 5^x, y 3^x, and y 1 10^x. a. Sketch graphs of the equations on one set of axes.

b. What points, if any, do the three graphs have in common?

c. In which graph does the y-value increase at the greatest rate as the x-value increases?

d. Use the graphs to figure out which of the equations is not an example of exponential growth. Explain how you know.

e. Use the equations to figure out which is not an example of exponential growth. Explain how you know.

Name ____________________________________________ Date ____________ Class ____________

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93

Name ____________________________________________ Date ____________ Class ____________

8. Consider the graphs of y 5 2^x, y 5 6^x and y 5 0.5x 1 1.

Which of the following statements are true?

Select all that apply.

n All three graphs are examples of exponential growth.

n The three graphs intersect at (0, 1).

n The graph of y 5 0.5x 1 1 does not intersect the x-axis.

n The graphs of y 5 2^x and y 5 6^x have only one point in common.

9. Suppose you deposit $5,000 in a savings account that earns interest of 3% per year on the current balance in the account.

Using the tiles, write an equation that represents the value of the account, V, after t years.

1.30

1.03

5,000

0.03

0.97

^t

V 5 ( )( )

10. Choose whether each table might represent a relationship that is linear, exponential, inverse, or none of these by circling the appropriate word or phrase.

a. _x y 4

1

4.1616 4.08

2

4.2448 3

4.3297 4

0 b.

x y

1 2 3 4 5

2 5

1 5

2 15

1 20

2 25

Q

^linearexponential inverse none of these

U

Q

^linearexponential

inverse none of these

U

c. _x y 250

1

390.63 312.5

2

488.28 3

610.35 4

0 d.

x y 2.1

1 7.1 4.6

2 9.6

3 12.1

4 0

Q

^linearexponential inverse none of these

U

Q

^linearexponential

inverse none of these

U

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Skill: Compound Interest

Growing, Growing, Growing For Exercises 1–2, complete each table. Compound the interest annually.

1. $5,000 at 6% for 4 years.

2. $7,200 at 3% for 4 years

3. Suppose one of your ancestors invested $500 in 1800 in an account paying 4% interest compounded annually. Write an exponential function to model the situation. Find the account balance in each of the following years.

a. 1850

b. 1900

c. 2000

d. 2100

Principal at Beginning

Interest Balance of Year

Year 1: $5,000 Year 2:

Year 3:

Year 4:

Principal at Beginning

Interest Balance of Year

Year 1: $7,200 Year 2:

Year 3:

Year 4:

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95

Name ____________________________________________ Date ____________ Class ____________

1. Joan and Jeff are standing 50 meters apart. They take turns walking toward each other. Jeff walks one half the distance between them, then Joan walks one half the distance between them. They take turns, each walking one half the remaining distance between them. Suppose that each walks 4 times (8 rounds) during this exercise.

a. Make a table showing how far apart Joan and Jeff are after each of the first 8 rounds.

b. Make a graph of your data from part (a).

c. Suppose that Joan and Jeff start over and take turns walking 3 feet toward each other. Make a table and a graph for this walking exercise showing how far apart they will be after each of the first 8 rounds.

d. Compare the tables and graphs for the two situations. Explain the similarities and the differences you see.

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Growing, Growing, Growing 2. A tree farm has begun to harvest a section of trees that was planted a number

of years ago.

a. Suppose the relationship between the year and the trees remaining is exponential. Approximate the decay factor for this relationship.

b. Write an equation for the relationship between time and trees remaining.

c. Evaluate your equation for each of the years shown in the table below to find the approximate number of trees remaining.

d. The owners of the farm intend to stop harvesting when only 15% of the trees remain. During which year will this occur? Explain your reasoning.

Year

Trees Remaining

10 15 20 25 30 35 40

Supply of Trees Year

Trees Remaining 0 10,000

1 9,502

2 9,026

3 8,574

4 8,145

5 7,737

6 7,350

7 6,892

8 6,543 Supply of Trees

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97

Name ____________________________________________ Date ____________ Class ____________

3. Kai’s brother collects fuzzy insects called tribetts. The tribett population decreases by 30% each year.

a. Complete the table to show the number of tribetts at the end of the first 5 years for a starting population of 10,000 tribetts.

b. Write an equation for the relationship between years and number of tribetts.

c. In what year will there first be fewer than 1,000 tribetts?

4. There are 64 volleyball teams entered in the state competition. In the first round of play, each team plays one other team, so 32 games will be played in the single elimination tournament. The winners from these games play each other in a second round. The winners of the second round play each other in a third round. This continues until there is a final winning team. There are no tie games; games are played into overtime if needed.

a. How many rounds of play are needed before a winner is determined?

Explain your reasoning.

b. How many total games are played before a winner is determined? Explain.

c. Suppose an additional round of play is added to the playoffs. How many teams would start in the playoffs? Explain.

Year Tribetts

0 1 2 3 4 5

Tribett Population

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Name ____________________________________________ Date ____________ Class ____________

7. The tortoise and the hare are in a 150-meter race. Suppose the tortoise can walk 3 meters in 1 minute. The hare moves in such a way that he hops __ ¹₃ of whatever distance remains each minute. Complete each statement by circling the appropriate number.

After 3 minutes, the tortoise and the hare will be approximately

Q 

⁹⁷¹⁰⁵¹³⁵

U

meters apart.

After

Q 

⁴¹⁰⁵⁰

U

minutes, the hare will be less than 3 meters from the finish line.

A graph of the distance the tortoise walks each minute is an example

of a(an)

Q

exponential linear

inverse

U

^model.

A graph of the distance the hare hops each minute is an example

of a(an)

Q

exponential linear

inverse

U

^model.

5. The population of a certain species of bird is decreasing by 10% each year. If the population in year 0 is 15,000, which of the following statements are true? Select all that apply.

n In year 7, there will be fewer than 7,000 of this species of bird.

n After 5 years, there will be

approximately 8,857 of this species of bird.

n The graph of this relationship shows exponential decay.

n The decay factor is 0.10.

n After 1 year, there will be 13,500 of this species of bird.

6. In some areas, home values have decreased over the past 10 years. The table shows the decrease in the home value of one house.

Year 0 1 2 3 4 5

$250,000

$242,500

$235,225

$228,168

$221,323

$214,684 Home Value

Using the tiles, write an equation that represents the value of the home, v, after t years.

1.03

0.03

250,000

0.97

1.30

^t

v 5 ( )( )

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99

Skill: Exponential Growth and Decay

1. Complete the table for integer values of x from 0 to 4. Then graph the function.

y ⴝ 50(0.2)^x

2. Suppose the acreage of forest is decreasing by 2% per year because of development. There are currently 4,500,000 acres of forest. Write an exponential function to model this situation and determine the amount of forest land after each of the following.

a. 3 years b. 5 years

c. 10 years d. 20 years

y

10 20 30

O x

2 4

40 50

6 8 10

x y (x, y)

0 1 2 3 4

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Skill: Exponential Growth and Decay (continued)

3. a. Write an exponential function to model this situation: A $10,500 investment has a 15% loss each year.

b. Find the value of the investment after each length of time.

1 year 2 years 4 years 10 years

4. a. Write an exponential function to model this situation: A city of 2,950,000 people has a 2.5% annual decrease in population.

b. Find the population of the city after each length of time.

5. Write an exponential function to model this situation: A $25,000 purchase decreases 12% in value per year.

b. Find the value of the purchase after each length of time.

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101

Investigation 2

1. In parts (a)–(f), write two equivalent expressions: First, write the expression in an equivalent form using exponents, then write the expression in standard form.

a. 2⁵ 2⁵ b. 4³ 2⁵

c. 25⁴ d.

e. 10² 2 5 f. 3³ 2³

2. In parts (a)–(d), find the units digit of the standard form of the expression.

a. 12¹⁰ b. 11²³

c. 23¹⁹ d. 17¹⁷

3⁴ 3

Name ____________________________________________ Date ____________ Class ____________

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Name ____________________________________________ Date ____________ Class ____________

3. Consider these three equations: y ⫽ 0.625^x, y ⫽ 0.375^x, and y ⫽ 1 ⫺ 0.5x.

a. Sketch graphs of the equations on one set of axes.

b. What points, if any, do the three graphs have in common?

c. In which graph do the y-values decrease at a faster and faster rate as the x-values increase?

4. Decide whether each statement is true or false. Explain your reasoning.

a. 3⁵⫹ 3⁵⫽ 3¹⁰ b. 5⁴⫹ 2⁴⫽ 7⁴

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Name ____________________________________________ Date ____________ Class ____________

5. In parts (a)–(f), use the properties of exponents to evaluate each expression.

a. b.

c. d.

e. f. (8^⫺14)(8¹⁴)

6. Write each number in standard form as a decimal.

a. 6 ⫻ 10¹ b. 2.1 ⫻ 10² c. 3.4 ⫻ 10³

d. 4 ⫻ 10^⫺1 e. 1.7 ⫻ 10^⫺2 f. 6.2 ⫻ 10^⫺3

2 5 7

2435

243

(25⁵⁴)

2

(18⁹)(₁₈¹) ⁹ 5

(41³⁷)(41⁴⁷) (36¹⁶)⁶

103

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7. Write each number in scientific notation.

a. 6,000,000 b. 27,000,000,000 c. 568,000,000

d. 0.0027418 e. 0.000030684 f. 0.06874916

g. 3⁶ h. ( )⁶ ^{i. 30}⁶

8. Rewrite each expression in scientific notation.

a. (4.0 10⁴) (2.4 10³)

b. (3.0 10⁸) (1.2 10⁶)

c. (4.0 10⁴) (2.4 10³)

d. (5.4 10⁵) (6.0 10⁴)

9. Without graphing these equations, describe and compare their graphs. Be as specific as you can.

y 5^x y 0.5^x

1 3

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105

Name ____________________________________________ Date ____________ Class ____________

Growing, Growing, Growing Investigation 1 Investigation 5

10. Which of the following expressions are equivalent to 5 ² ? 5 ³ ? Select all that apply.

n 5 ? 5 ? 5 ? 5 ? 5 n 5 ⁶

n 5 ^___ 5 ¹⁰⁵ n ( 5 ³) ² n ( 5 ²) ² ? 5

11. Compare each pair of expressions by circling the appropriate symbol to make each statement true.

a. 2 ⁶ Q^,^.⁵U⁽²²⁾⁽²⁴⁾

b. 81 Q^,^.⁵U⁽⁹²⁾⁽⁹²⁾

c. ( 4 ³ ) ( 4 ² ) Q^,^.⁵U^{1 6}²

d. 1 6 ^__³² Q^,^.⁵U⁸²

e. ( 5 ¹^__⁴) ⁷ Q^,^.⁵U⁵⁴

12. Match each number in the left-hand column with an equivalent expression in the right-hand column.

12,000,000 (2.4 3 1 0 ⁴ ) 3 (5.0 3 1 0 ²⁵ )

1.2 1.2 3 1 0 ⁷

0.00000012 (1.2 3 1 0 ²¹ ) 4 (1.0 3 1 0 ² )

1,200,000 1.2 3 10 ²⁷

0.0012 (3.6 3 1 0 ⁴ ) 4 (3.0 3 1 0 ²² )

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Skill: Simplifying Exponential Expressions

Find an equivalent expression.

1. 3²⭈ 3⁵

2. 1³⭈ 1⁴

3. 5⁴⭈ 5³

4. 4.5⁸⭈ 4.5²

5. 3³⭈ 3 ⭈ 3⁴

Replace each ⵧ^{with ⴝ,}^{R, or S.}

6. 3⁸ⵧ^{3 ⭈ 3}⁷ ^{7. 49}ⵧ⁷²^{⭈ 7}² ^{8. 5}³^{⭈ 5}⁴ⵧ²⁵²

Simplify each expression.

9.

10.

11.

12.

13. ⁽²³⁾

5

(23)⁸ 7⁵ 7³ (24)⁸ (24)⁴ 8⁴ 8⁰ (23)⁶ (23)⁸