1. Modeling with Proportions and Variation
Section 6.6
Modeling with Proportions and Variation

2. Objectives Proportions Direct Variation Inverse Variation
Joint Variation

3. Proportions
A proportion is a statement (equation) that two ratios (fractions) are equal. The following property is a convenient way to solve proportions:
is equivalent to
provided b ≠ 0 and d ≠ 0.

4. 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? Solution Let x be the equivalent amount of calories.
Thus, in 30 minutes, Francis will burn 444 calories.

5. 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. Solution The triangles are similar because the measures of its corresponding angles are equal. Therefore corresponding sides are proportional.
6 ft h
8 ft
32 ft
The tree is 24 feet tall.

6. 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
Total # tagged
Tagged in sample
Total # in sample
Total # of birds in forest
To the nearest hundred, there are approximately 2900 birds in the forest.

9. Example
Let y be directly proportional to x, or vary directly with x. Suppose y = 9 when x = 6. Find y when x = 13. Solution Step 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.

10. 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

11. 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.
y = 23x,
y = 23(19)
19 hours of work would pay $437.00
y = 437

13. Example
Let y be inversely proportional to x, or vary inversely with x. Suppose y = 6 when x = 4. Find y when x = 8. Solution Step 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.

14. 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.
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. x 3 7 9 12 y 28 32 48
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. x 5 10 12 15 y 6 4
The equation y = 6x models the data. The data represents direct variation. x 8 11 14 21 y 48 66 84
126

15. JOINT VARIATION
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

16. 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? Solution The strength of the beam is modeled by S = kwt2.

17. 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