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Date: 5-7-12 Essential Question: What is a direct variation? Topic: 8-9 Direct Variation Objective To use direct variation to solve problems. The table below shows the mass m of a gold bar whose volume is V cubic centimeters. Volume in cubic Mass in centimeters: V grams: m Direct Variation 1 19.3 2 3 38.6 57.9 4 5 77.2 96.5 You can see that . This equation defines a linear function. Note that if the volume of the bar is doubled, the mass is doubled; if the volume is tripled, the mass is tripled, and so on. You can say that the mass varies directly as the volume. This function is an example of a direct variation. A direct variation is a function defined by an equation of the form 𝑦 𝑘𝑥 where k is a nonzero constant. You can say that y varies directly as x. The constant k is called the constant of variation. Summary 1 Example 1 State whether or not each equation defines a direct variation. For each direct variation, state the constant of variation. If not a direct variation, explain why. a. Yes constant of variation = 3 b. Yes constant of variation = 5 c. No Domain variable (r) is squared. d. No y-intercept = 2 2 Exercise 1 State whether or not each equation defines a direct variation. For each direct variation, state the constant of variation. If not a direct variation, explain why. a. b. c. d. e. 3 Example 2 Given that m varies directly as n and that m = 42 when n = 2, find the following: a. the constant of variation Let . Substitute m = 42 and n = 2 b. the value of m when n = 3 Let . Substitute k = 21 and n = 3 c. the value of n when m = 7 Let . Substitute k = 21 and m = 7 4 Exercise 2 Given that v varies directly as u and that v = 12 when u = 6, find the following: a. the constant of variation b. the value of v when u = 3 c. the value of u when v = 4 5 Constant of Proportionality ) and ( ) are two ordered pairs of a direct Suppose ( variation defined by and that neither x1 nor x2 is zero. Since ( ) and ( ) must satisfy , you know that From these equations you can write the ratios Since each ratio equals k. the ratios are equal. This equation, which states that two ratios are equal, is a proportion. For this reason, k is sometimes called the constant of proportionality, and y is said to be directly proportional to x. When you use a proportion to solve a problem, you will find it helpful to recall that the product of the extremes equals the product of the means. (See lesson 7-2.) In the proportion a:b = c:d, a and d are called the extremes, and b and c are called the means. You can use the multiplication property of equality to show that in any proportion the product of the extremes equals the product of the means. That is: 𝑎 If 𝑏 𝑐 𝑑 , then ad = bc. 6 The amount of interest earned on savings is directly proportional to the amount of money saved. If $104 interest is earned on $1300, how much interest will be earned on $1800 in the same period of time? Example 3 Step 1 The problem asks for the interest on $1800 if the interest on $1300 is $104. Step 2 Let i, in dollars, be the interest on d dollars. i1 = 104 i2 = ? . d1 = 1300 d2 = 1800 Step 3 An equation can be written in the form Step 4 Step 5 the interest earned on $1800 will be $144. 7 Example 3 Solution 2 To solve Example 3 by the method shown in Example 2, first write the equation . Then solve for the constant of variation, k, by using the fact that when . Use the value of k to find the value of i when . Exercise 3 A certain car uses 15 gal of gasoline in 3 h. If the rate of gasoline consumption is constant, how much gasoline will the car use on a 35-hour trip? 8 Exercise 3 ct’d A restaurant buys 20 lb of ground beef to prepare 110 servings of chili. At this rate, how many servings can be made with 30 lb of ground beef? Homework P 393 Written Exercises: 1-21 every 4th P 395 Problems: 1, 5, 9 9