6/22/ Ratio, Proportion and Variation 1 Starter An HMO pamphlet contains the following recommended weight for women: Give yourself 100 pounds for the first 5 feet plus 5 pounds for every inch over 5 feet tall.” Using this description, which height corresponds to an ideal weight of 135 pounds? (i) = 135 5i = 35 i = 7 inches

6/22/ Ratio, Proportion and Variation 2 Today’s Agenda 1. Review Homework 2. Study the new lesson 6.4 Ratio, Proportion and Variation In this chapter we are learning to use formulas and mathematical models in new ways. With these skills, we are gaining insights into a variety of issues, ranging from the growing diversity of the U.S. population to concerns about Social Security and even the positive benefits that humor and laughter can have on our lives! 3. Complete the next assignment

6/22/ Ratio, Proportion and Variation 3 Review Homework Applications of Linear Equations Beginning on page 18

6/22/ Ratio, Proportion and Variation 4 Ratio, Proportion, and Variation Objectives 1. Solve proportions. 2. Solve problems using proportions. 3. Solve direct variation problems. 4. Solve inverse variation problems.

6/22/ Ratio, Proportion and Variation 5 Proportions  Ratio compares quantities by division. Example: a group contains 60 women and 30 men. The ratio of women to men is:  Proportion is a statement that says two ratios are equal:

6/22/ Ratio, Proportion and Variation 6 Example 1 The Cross-Products Principle for Proportions If a = c then ad = bc. (b  0 and d  0) b d The cross products ad and bc are equal. Solve this proportion for x:

6/22/ Ratio, Proportion and Variation 7 Applications of Proportions Solving Applied Problems Using Proportions 1. Read the problem and represent the unknown quantity by x ( or any letter ). 2. Set up a proportion by listing the given ratio on one side and the ratio with the unknown quantity on the other side. Each respective quantity should occupy the same corresponding position on each side of the proportion. 3. Drop units and apply the cross-products principle. 4. Solve for x and answer the question.

6/22/ Ratio, Proportion and Variation 8 Example 2 Calculating Taxes The property tax on a house whose assessed value is $65,000 is $825. Determine the property tax on a house with an assessed value of $180,000, assuming the same tax rate. Solution: Step 1. Let x = tax on a $180,000 house. Step 2. Set up the proportion:

6/22/ Ratio, Proportion and Variation 9 Example 2 continued Step 3 Drop the units and apply the cross products principle, Step 4 Solve for x and answer the question. The property tax on the $180,000 house is approximately $

6/22/ Ratio, Proportion and Variation 10 Example 3 Direct Variation As one quantity increases, the other quantity increases and vice versa.  An alligator’s tail length varies directly as its body length. An alligator with a body length of 4 feet has a tail length of 3.6 feet. What is the tail length of an alligator whose body length is 6 ft?

6/22/ Ratio, Proportion and Variation 11 Example 3 continued Solution Step 1. Let x = tail length of an alligator whose body length is 6 feet. Step 2. Set up the proportion:

6/22/ Ratio, Proportion and Variation 12 Example 3 continued Direct Variation Step 3. Apply the cross-products principle, solve and answer the question. An alligator whose body length is 6 feet has a tail length measuring 5.4 feet.

6/22/ Ratio, Proportion and Variation 13 Inverse Variation  As one quantity increases, the other quantity decreases and vice versa.  Setting up a proportion when y varies inversely as x The first value for y = The second value for y The value for x corresponding to the second value for y to the first value for y NOTE: In an inverse variation situation, corresponding values are not placed in the same ratio. Placing corresponding values in opposite ratios allows one quantity to increase while the other decreases.

6/22/ Ratio, Proportion and Variation 14 A bicyclist tips the cycle when making a turn. The angle B, formed by the vertical direction and the bicycle, is called the banking angle. The banking angle varies inversely as the cycle’s turning radius. When the turning radius is 4 feet the Banking angle is 28 . What is the banking angle when the turning radius is 3.5 feet? Example 4 Inverse Variation

Step 3 and 4. Apply the cross products principle, solve, and answer the question. 4(28) = 3.5x 112 = 3.5x 32 = x When the turning radius is 3.5 feet, the banking angle is 32 . Example 4 continued Solution Step 1. Represent the unknown x. x = banking angle when turning radius is 3.5 feet. Step 2. Set up the proportion. When the turning radius is 4 feet the Banking angle is 28 . What is the banking angle when the turning radius is 3.5 feet? The first value for y = The second value for y The value for x corresponding to the second value for y to the first value for y

6/22/ Wrap Up In a hurricane, the wind pressure varies directly as the square of the wind velocity. If wind pressure is a measure of a hurricane’s destructive capacity, what happens to this destructive power when the wind velocity doubles? 6.4 Ratio, Proportion and Variation Therefore, if the wind velocity doubles, the pressure quadruples.

6/22/ Assignment  Assignment beginning on page 338: #’s 14, 15, 25, 29, 31, 34, 33, 35, 37, 39, 40 & 42  Alternate Assignment: 6.4 Ratio, Proportion and Variation 6.4 Ratio, Proportion and Variation