Identify Ratios, Rates, and Proportions
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1 Math 35 "Fall 08" 6.9 "Proportion and Variation" Objectives: * Identify ratios, rates, and proportions * Solve proportions * Use proportions to solve problems * Solve problems involving - Similar triangles - Direct variation - Inverse variation - Joint variation - Combined variation Preliminaries: In this section, we will discuss the ratio-proportion model and four variation model. They can be used to solve a variety of application problems. Identify Ratios, Rates, and Proportions The quotient of two numbers or two quantities with the same units is often called a ratio. For example: When we compare two quantities having di erent units, we call the comparison a rate. For example: Page: 1
2 An equation indicating that two ratios or rates are equal is called a proportion. For example: The Fundamental Property of Proportions In a proportion, the product of the extreme is equal to the product of the means Solve Proportions To solve a proportion, we can use the fundamental property of proportions Example 1: (Solving proportions) Solve each proportion: a) x + 3 x = x x + 6 b) 3x = x x + 2 Page: 2
3 Use Proportions to Solve Problems We can use proportions to solve many application problems. Example 2: (Using proportions to solve problems) To make a dessert, a chef needs to purchase 14 pears. If they are on sale at 6 for $2:34, what will the cost of 14 pears? Large Solve Problems Involving Similar Triangles Similar Triangles If two triangles are similar, then 1: The three angles of the rst triangle have the same measure, respectively, as the three angles of the second triangle. 2: The lengths of all corresponding sides are in proportion. Example 3: (Solving problems involving similar triangles) A tree casts a shadow of 29 feet at the same time as a vertical yardstick casts a shadow of 2:5 feet. Find the height of the tree. Page: 3
4 Solve Problems Involving Direct Variation Direct Variation The words "y varies directly as x" or "y is directly proportional to x" means that: for some nonzero constant k. The constant k is called the constant of variation or the constant of proportionality Solving Variation Problems To solve a variation problem: 1: Translate the verbal model into an equation 2: Substitute the rst set of values into the equation from step 1 to determine the value of k 3: Substitute the value of k into the equation from step1 4: Substitute the remaining set of values into the equation from step 3 and solve for the unknown Example 4: (Solving problems involving direct variation) The force of gravity acting on an object varies directly as the mass of the object. The force on a mass of 5 kilograms is 49 newtons. What is the force acting on mass of 12 kilograms? Solve Problems Involving Inverse Variation Inverse Variation The words "y varies inversely as x" or "y is inversely proportional to x" means that: for some nonzero constant k. The constant k is called the constant of variation. Page: 4
5 Example 5: (Solving problems involving inverse variation) The intensity I of light received from a light source varies inversely as the square of the distance from the light source. If a photographer, 16 feet away from his subject, has a light meter reading of 4 footcandles of luminance, what will the meter read if the photographer moves in for a close-up 4 feet away from the subject? Solve Problems Involving Joint Variation Joint Variation If y varies jointly with x and z, then The nonzero constant k is called the constant of variation. Example 6: (Solving problems involving joint variation) The force of the wind on a billboard varies jointly as the area of the billboard and the square of the wind velocity. When the wind is blowing at 20 mph, the force on a billboard 30 feet wide and 18 feet high is 972 pounds. Find the force on a billboard having an area of 300 square feet caused by a 40 mph wind. Page: 5
6 Solve Problems Involving Combined Variation Many applied problems involve a combination of direct and inverse variation. Such variation is called combined variation. Example 7: (Solving problems involving combined variation) The time it takes to build a highway varies directly as the length of the road, and inversely as the number of workers. If it takes 100 workers 4 weeks to build 2 miles of highway, how long will it take 80 workers to build 10 miles of highway? Page: 6
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