Copyright © 2013, 2009, 2005 Pearson Education, Inc. Section 6.6 Modeling with Proportions and...

Copyright © 2013, 2009, 2005 Pearson Education, Inc.

Section 6.6

Modeling with Proportions and

Variation


Objectives

• Proportions

• Direct Variation

• Inverse Variation

• Joint Variation


Proportions

A proportion is a statement (equation) that two ratios (fractions) are equal.

The following property is a convenient way to solve proportions:

a

b d

c is equivalent to ,ad bc

provided b ≠ 0 and d ≠ 0.


Example

On an elliptical machine, Francis can burn 370 calories in 25 minutes. If he increases his work time to 30 minutes, how many calories will he burn?SolutionLet x be the equivalent amount of calories.

25 30

370 x

Minutes Minutes=

Calories Calories

25 11,100x

444x Thus, in 30 minutes, Francis will burn 444 calories.


Example

A 6-foot tall person casts a shadow that is 8-foot long. If a nearby tree casts a 32-foot long shadow, estimate the height of the tree.SolutionThe triangles are similar because the measures of its corresponding angles are equal. Therefore corresponding sides are proportional.

6 ft h

8 ft 32 ft

32

6 8

h

Height Shadow length=

Height Shadow length8 192h

24h The tree is 24 feet tall.


Example

A forest ranger catches 616 birds, tags them, and releases them. Two months later, 244 birds are captured and 51 were found with tags. Estimate how many birds are in the forest to the nearest hundred. Solution

51 616

244 B

51 150,304B

2947B To the nearest hundred, there are approximately 2900 birds in the forest.

Tagged in sample

Total # in sample

Total # tagged

Total # of birds in forest


Example

Let y be directly proportional to x, or vary directly with x. Suppose y = 9 when x = 6. Find y when x = 13.SolutionStep 1 The general equation is y = kx. Step 2 Substitute 9 for y and 6 for x in y = kx. Solve for k.Step 3 Replace k with 9/6 in the equation y = 9x/6.Step 4 To find y, let x = 13.

9 6

9

6

y kx

k

k

9

69

(13)619.5

y x

y

y


Example

The table lists the amount of pay for various hours worked.

a. Find the constant of proportionality. b. Predict the pay for 19 hours of work.

Hours Pay

6 $138

11 $253

15 $345

23 $529

31 $713

6, 138

11, 253

15, 345

23, 529

31, 713

0

100

200

300

400

500

600

700

800

0 10 20 30 40P

ay

(do

llars

)

Hours


Example (cont)

The slope of the line equals the proportionality, k. If we use the first and last data points (6, 138) and (31, 713), the slope is

The amount of pay per hour is $23. The graph of the line y = 23x, models the given graph.To find the pay for 19 hours, substitute 19 for x.

713 138

31 6k

23

y = 23x,y = 23(19)y = 437

19 hours of work would pay $437.00


Example

Let y be inversely proportional to x, or vary inversely with x. Suppose y = 6 when x = 4. Find y when x = 8.SolutionStep 1 The general equation is y = k/x. Step 2 Substitute 6 for y and 4 for x in Solve for k.Step 3 Replace k with 24 in the equation y = k/x.Step 4 To find y, let x = 8.

64

24

ky

xk

k

24

83

ky

x

y

y


Example

Determine whether the data in each table represent direct variation, inverse variation, or neither. For direct and inverse variation, find the equation.a.

b.

c.

x 3 7 9 12

y 12 28 32 48

x 5 10 12 15

y 12 6 5 4

x 8 11 14 21

y 48 66 84 126

Neither the product xy nor the ratio y/x are constant in the data in the table. Therefore there is neither direct variation nor indirect variation in this table.

As x increases, y decreases. Because xy = 60 for each data point, the equation y = 60/x models the data. This represents an inverse variation.

The equation y = 6x models the data. The data represents direct variation.


, 0 and 1,xf x a a a

Let x, y, and z denote three quantities. Then z varies jointly with x and y if there is a nonzero number k such that

JOINT VARIATION

.z kxy


Example

The strength S of a rectangular beam varies jointly as its width w and the square of its thickness t. If a beam 5 inches wide and 2 inches thick supports 280 pounds, how much can a similar beam 4 inches wide and 3 inches thick support?SolutionThe strength of the beam is modeled by S = kwt2.

2280 5 2k

280

5 4k

14k


Example (cont)

Thus S = 14wt2 models the strength of this type of beam. When w = 4 and t = 3, the beam can support

S = 14 ∙ 4 ∙ 32 = 504 pounds

Copyright © 2013, 2009, 2005 Pearson Education, Inc. Section 6.6 Modeling with Proportions and...

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