7 The Mathematics of Patterns & Nature

The Mathematics of Patterns & Nature

Linear Patterns Recognize and describe a linear pattern.

Use a linear pattern to predict a future event.

Recognize a proportional pattern.

Exponential Patterns Recognize and describe an exponential pattern.

Use an exponential pattern to predict a future event.

Compare exponential and logistic growth.

Quadratic Patterns Recognize and describe a quadratic pattern.

Use a quadratic pattern to predict a future event.

Compare linear, quadratic, and exponential growth.

Fibonacci & Other Patterns Recognize and describe the Fibonacci pattern.

Analyze geometric Fibonacci patterns.

Recognize and describe other patterns in mathematics.

7 7.1

7.2

7.3

7.4

Copyright © Larson Texts, Inc. All rights reserved.

Seal Call Success

Pro

bab

ility

of

succ

ess

Mother’s call rate (calls/min)

50 10 15 20

0.2

0.4

0.6

0.8

1.0

1.1

0.1

0.0

0.3

0.5

0.7

0.9

Pup’s responseCall moveCall onlyMove onlyNone

Pup’s responseCall move

Example 6 on page 317 compares a mother seal’s call rate with the probability that the mother will relocate her baby after going to sea. What is the mother’s and baby’s best strategy?


302 Chapter 7 The Mathematics of Patterns & Nature

Recognize and describe a linear pattern.

Use a linear pattern to predict a future event.

Recognize a proportional pattern.

Recognizing a Linear Pattern

A sequence of numbers has a linear pattern when each successive number increases (or decreases) by the same amount.


Anthropologists use tables like those at the left to estimate the height of a person based on part of the person’s skeleton.

a. Does the table relating the length of a man’s femur (upper leg bone) to the man’s height represent a linear pattern?

b. The femur length of a Roman soldier is 18 inches. What was the height of the Roman soldier?

SOLUTION

a. To determine whether the table represents a linear pattern, fi nd the differences between consecutive terms.

Each time the femur length increases by 1 inch, the height of the man increases by 1.88 inches. So, the pattern is linear.

b. From the table, an 18-inch femur corresponds to a height of about 66 inches. In other words, the Roman soldier was about 5' 6".

Checkpoint Help at

For women, femur length and height are related as follows.

Height in inches = 1.95(femur length) + 28.7

c. Use a spreadsheet to make a table for this formula.

d. Use the spreadsheet to graph the data in the table and verify that the points on the graph lie on a line.

7.1 Linear Patterns

Study TipLinear patterns involving two variables are called linear because when one variable is graphed in relationship to the other variable, the result is a line.

Femur length (inches)

Hei

ght (

inch

es)

Men’s Height

1510 20 25

55

50

60

65

70

75

80

AFemur

14151617

BHeight

Length (in.) (in.)58.3260.2062.0863.96

18 65.8419 67.7220 69.6021 71.4822 73.3623 75.2424 77.12

21

3456789

101112

19 67.7220 69.6021 71.4822 73.3623 75.2424 77.12

789

101112

AFemur

14151617

BHeight

Length (in.) (in.)58.3260.2062.0863.96

18 65.848 19 67 72

123456

1.881.881.881.88

betwee

Le1


http://math.andyou.com/content/07/01/data/mthu_data_0701_302_01.xls

7.1 Linear Patterns 303


The table relates a man’s shoe size to the length of his foot.

To measure your foot, trace it on a piece of paper. Mark the front and back of your foot. Then measure the length.

a. Does the table represent a linear pattern? Explain.

b. Use a spreadsheet to graph the data. Is the graph linear?

SOLUTION

a. To determine whether the table represents a linear pattern, fi nd the differences between consecutive terms.

Notice that each time the foot length increases by 0.17 (about 1/6) inch, the shoe size increases by a half size. So, the pattern is linear.

b. The points on the graph do lie on a line. So, the graph is linear.

Checkpoint Help at

Use the table at Math.andYou.com that relates a woman’s shoe size to the length of her foot.

c. Does the table represent a linear pattern? Explain.

d. Use a spreadsheet to graph the data. Is the graph linear?

AFoot

9.309.479.64

B

Length Shoe(inches) Size

6.06.57.0

9.81 7.59.98 8.0

10.15 8.510.32 9.010.49 9.510.66 10.010.83 10.511.00 11.011.17 11.511.34 12.011.51 12.511.68 13.011.85 13.512.02 14.012.19 14.512.36 15.0

123456789101112131415161718192021 A

Foot

9.309.479.64

B

Length Shoe(inches) Size

6.06.57.0

9.81 7.59.98 8.0

9 10 15 8 5

123456

0.170.170.170.17

Foot length (inches)

Sho

e si

zeU.S. Men’s Shoe Size

98 10 11 12 13

6

4

8

10

12

14

16

differences

L




Using a Linear Pattern to Predict a Future Event

One common use of linear patterns is predicting future events.

Predicting a Future Event

The graph shows the ages of American women at the time of their fi rst marriage from 1960 through 2010. Use the graph to predict the age in 2020.

SOLUTION

The pattern looks roughly linear. One way to estimate the age in 2020 is to draw a “best-fi tting line” to approximate the data. Then use the line to estimate the age in 2020.

From the graph, it appears that the age of women at their fi rst marriage in 2020 will be about 27.5.

Checkpoint Help at

Use the graph below to predict the marriage rate for women in the United States in 2020. How do these data relate to the data in Example 3?Study Tip

The procedure described in Example 3 is called linear regression. It is a commonly used procedure in statistics.

Ages of U.S. Women at First Marriage

Ag

eYear

1950 1960 1970 1980 1990 2000 2010 2020

20

0

22

24

26

28

Ag

e

Year

1950 1960 1970 1980 1990 2000 2010 2020

20

0

22

24

26

28

In 2020, theage will beabout 27.5.

Ages of U.S. Women at First Marriage

1960 1970 1980 1990 2000 2010 20200

20

40

60

80

100

Marriage Rate for U.S. Women

Rat

e p

er 1

000

un

mar

ried

wo

men

Year

28




Describing a Linear Pattern in Nature

The fi gure shows the annual northward migration of Canada Geese. Describe any linear patterns that you see in the fi gure.

SOLUTION

Through the central United States, the migration appears to be moving north at a rate of about 150 miles every 10 days.

Checkpoint Help at

The longest known migration of a mammal is that of the gray whale. It travels the 6000 miles between Baja California, Mexico, and the Bering Sea each spring and fall. Traveling at a rate of 4 miles per hour, how long does it take a gray whale to migrate each spring?

In this book, Serge Dedina discusses the conservation of the gray whale in Baja California, Mexico.

Feb 10 Feb 20 Mar 1 Mar 10 Mar 20 Mar 30 Apr 10 Apr 20 Apr 30

0 mi 130 mi 270 mi 420 mi 630 mi 760 mi 900 mi 1040 mi 1200 mi

Migration line

200 mi

March 1

March 10

March 20

March 30

April 10

April 20

April 30

Feb. 20Feb. 10

SOLUTION

Mi

March 1March 111 Feb. 20200Feb. 10

In this book Serge Dedina



Recognizing a Proportional Pattern

A pattern with two variables is proportional when one of the variables is a constant multiple of the other variable. Proportional patterns are also linear.


You hang different weights from a spring. You then measure the distance the spring stretches.

a. Describe the pattern. Is the distance the spring stretches proportional to the weight?

b. How much does the spring stretch when you hang 6 pounds from it?

SOLUTION

a. You can see that the distance the spring stretches is 3/2 times the weight in pounds.

3 —

2 × 0 = 0

3 —

2 × 1 =

3 —

2

3 —

2 × 2 = 3

3 —

2 × 3 =

9 —

2

3 —

2 × 4 = 6

So, the distance the spring stretches is proportional to the weight.

b. When you hang 6 pounds from the spring, it will stretch (3/2 × 6), or 9 inches.

Checkpoint Help at

The distance that a spring stretches depends on its elasticity. Data for a different spring are shown in the table.

c. Is this spring more or less elastic than the spring in Example 5? Explain.

d. How much will this spring stretch when you hang 7 pounds from it?

Weight(pounds)

Distancestretched (inches)

0 0

1 3 — 4

2 3 — 2

3 9 — 4

4 3

Throughout human history, most discoveries have occurred because humans observed patterns. From the patterns, they then formulated laws of nature. The law illustrated in Example 5 is called Hooke’s Law, after the English scientist Robert Hooke. The law states that the distance a spring stretches is proportional to the weight hanging on the spring.

01

23

45

6IN

CH

6 in.4 in.3 in.0 in.0 in. 32

92

Spring Stretching

Dis

tan

ce s

tret

ched

(in

ches

)

Weight (pounds)

0 1 2 3 4 5 6

1

0

2

4

3

5

6

7




Is the following statement true? Explain your reasoning.

As a human grows, its skull height is proportional to its total height.

SOLUTION

This is not true of humans. A baby’s skull height is about one-fourth of its total height. An adult’s skull height is only about one-eighth of its total height.

In the graph, notice that the total height is not a constant multiple of the skull height.

Checkpoint Help at

Baby reptiles are miniature versions of the adults. So, for reptiles, it istrue that “as a reptile grows, its skull length is proportional to its

total length.” Some horned lizards can grow up to a length of 8 inches. Use the photo to estimate the ratio of the lizard’s

skull length to its total length.

Most adult humans are between 6.5 and 8 heads tall. Comic action heros, however, are often 9 heads tall.

9 in. 9 in. 7.5 in. 7 in. 6.5 in. 6 in.

Adult

Adult8 heads at 9 in.

15 years7.5 heads at 9 in.

10 years7 heads at 7.5 in.

5 years6 heads at 7 in.

3 years5 heads at 6.5 in.

1 year4 heads at 6 in.

15years

10years

5years

3years

1year

Human Body Proportions

Tota

l hei

gh

t (i

nch

es)

Skull height (inches)

0 2 4 6 8 100

20

40

60

80

Proportional lineof 1-to-8 ratio



7.1 ExercisesFreshwater The table shows the pressures at various depths of freshwater. In Exercises 1– 4, use the table. (See Examples 1 and 2.)

1. Does the table relating depth and pressure represent a linear pattern? Explain your reasoning.

2. Use a spreadsheet to graph the data. Is the graph linear?

3. How much does the pressure increase for every foot of depth? Explain your reasoning.

4. Write a formula that relates the depth in feet to the pressure in pounds per square inch.

Seawater For seawater, depth and pressure are related as follows.

Pressure in pounds per square inch = 0.445(depth in feet) + 14.7

In Exercises 5–8, use this formula. (See Examples 1 and 2.)

5. Use a spreadsheet to make a table for the formula. Then graph the data and verify that the points on the graph lie on a line.

6. The recreational diving limit for a scuba diver is 130 feet. Find the pressure at this depth.

7. The wreck of the Lusitania lies about 300 feet beneath the Celtic Sea. Find the pressure at this depth.

8. The wreck of the Titanic lies about 12,500 feet beneath the Atlantic Ocean. Find the pressure at this depth.

The stern of the Titanic, pictured above, was crushed by water pressure as it sank to the bottom of the ocean.

Depth (feet)

Pressure (pounds per square inch)

0 14.70

10 19.03

20 23.36

30 27.69

40 32.02

50 36.35

60 40.68

70 45.01

80 49.34

90 53.67

100 58.00nd

eltic Sea.

tlantic Ocean.

100 58.00





Tree Growth The fi gure shows the circumference of a tree over a 4-year period. In Exercises 9 and 10, use the fi gure. (See Example 3.)

9. Make a scatter plot of the data. Use the scatter plot to estimate the circumference of the tree in 2011.

10. Find the diameter of the tree for each year. Make a scatter plot showing the diameter for each year. Use the scatter plot to predict the diameter of the tree in 2011.

(The formula for the diameter of a circle is d = C

— π

.) 11. Black-and-White Warbler The fi gure

shows the migration of the Black-and-white Warbler. Describe any linear patterns that you see in the fi gure. (See Example 4.)

12. Distance Traveled A Black-and-white Warbler fl ies about 20 miles per day across the United States during migration. Complete the table to estimate the distance that the Black-and-white Warbler travels between March 30 and each date. (See Example 4.)

Day Distance

March 30 0

March 31

April 2

April 6

April 13

April 21

May 1

2006: 18.85 in.

2007: 20.42 in.

2008: 21.99 in.

2009: 23.56 in.

2010: 25.13 in.

March 10

Migration line

March 20

March 30

April 10April 20

May 1May 10

May 20

May 30

March 20

March 10




Absorbance The fi gure shows light passing through a glass container that contains a substance. Absorbance is a unitless measure of the amount of light that a substance absorbs as light passes through it. In Exercises 13–16, (a) describe the pattern of the absorbance values, (b) make a scatter plot of the data, and (c) predict the next absorbance value in the pattern. (See Examples 5 and 6.)

13. The table shows the absorbance of light with a wavelength of 630 nanometers for a solution of the food dye Blue No. 1 in a 1-centimeter glass container at various concentrations.

14. The table shows the absorbance of light with a wavelength of 625 nanometers for a solution of the food dye Green No. 3 in a 1-centimeter glass container at various concentrations.

15. The table shows the absorbance of light with a wavelength of 527 nanometers for a solution of the food dye Red No. 3 with a concentration of 4 milligrams per liter in glass containers of various path lengths.

16. The table shows the absorbance of light with a wavelength of 500 nanometers for a solution of the food dye Red No. 40 with a concentration of 5 milligrams per liter in glass containers of various path lengths.

17. Yellow No. 5 The absorbance of light with a wavelength of 428 nanometers for a solution of the food dye Yellow No. 5 in a 1-centimeter glass container is proportional to the concentration of Yellow No. 5. The absorbance of a solution with a concentration of 2 milligrams per liter is 0.106. What is the absorbance of a solution with a concentration of 5 milligrams per liter? (See Examples 5 and 6.)

18. Yellow No. 6 The absorbance of light with a wavelength of 484 nanometers for a solution of the food dye Yellow No. 6 with a concentration of 6 milligrams per liter is proportional to the path length. The absorbance of the solution in a 0.5-centimeter glass container is 0.162. What is the absorbance of the solution in a 1.1-centimeter glass container? (See Examples 5 and 6.)

ss

Path length

Light in Light out

Absorption takesplace here.

Concentration (mg/L) Absorbance

0 0

1 0.164

2 0.328

3 0.492

4 0.656

Concentration (mg/L) Absorbance

0 0

2 0.312

4 0.624

6 0.936

8 1.248

Path length (cm) Absorbance

0 0

0.2 0.088

0.4 0.176

0.6 0.264

0.8 0.352

Path length (cm) Absorbance

0 0

0.3 0.078

0.6 0.156

0.9 0.234

1.2 0.312






Extending ConceptsLinear Regression in Excel In Exercises 19 and 20, use the information below.

You can use Excel to fi nd the best-fi tting line for a data set. Enter the data into a spreadsheet. Make a scatter plot of the data. Click on the scatter plot. From the chart menu, choose “Add Trendline.” Click on the “Options” tab. Check the box labeled “Display equation on chart.” Click “OK.” This will add the best-fi tting line and its equation to your scatter plot.

19. The data set relates the number of chirps per second for striped ground crickets and the temperature in degrees Fahrenheit.

(Source: George W. Pierce, The Song of Insects, Harvard University Press, 1948)

a. Enter the data into a spreadsheet and make a scatter plot of the data.

b. Graph the best-fi tting line on your scatter plot and fi nd its equation.

c. Estimate the temperature when there are 19 chirps per second.

d. Estimate the temperature when there are 22 chirps per second.

20. Data were collected from a sample of 414 infants, grouped by month of birth. The data set relates the average monthly temperature (in degrees Fahrenheit) 6 months after the infants were born and the average age (in weeks) at which the infants learned to crawl.

(Source: Janette Benson, Infant Behavior and Development, 1993)

a. Enter the data into a spreadsheet and make a scatter plot of the data.

b. Graph the best-fi tting line on your scatter plot and fi nd its equation.

c. Estimate the average crawling age for infants when the average temperature 6 months after they are born is 55°F.

d. Estimate the average crawling age for infants when the average temperature 6 months after they are born is 475°F. Is this temperature reasonable? Is your estimate reasonable? Explain your reasoning.

Chirps per second Temperature (°F)

20.0 88.6

16.0 71.6

19.8 93.3

18.4 84.3

17.1 80.6

15.5 75.2

14.7 69.7

17.1 82.0

Chirps per second Temperature (°F)

15.4 69.4

16.2 83.3

15.0 79.6

17.2 82.6

16.0 80.6

17.0 83.5

14.4 76.3

(S J B I f B h i d D

Average temperature (°F)

Average crawlingage (in weeks)

66 29.84

73 30.52

72 29.70

63 31.84

52 28.58

39 31.44

l 1993)

Average temperature (°F)

Average crawlingage (in weeks)

33 33.64

30 32.82

33 33.83

37 33.35

48 33.38

57 32.32

y Press, 1948)

e data.

uation.

nd.

ond.

by month of birth. egrees Fahrenheit) 6 months

) at which the infants learned to crawl.

80.6

83.5

76.3

e Average crawlingage (in weeks)

64



7 The Mathematics of Patterns & Nature

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