Presentation on theme: "Modeling with Proportions and Variation"— Presentation transcript:
1Modeling with Proportions and Variation Section 6.6Modeling with Proportions and Variation
2Objectives Proportions Direct Variation Inverse Variation Joint Variation
3ProportionsA proportion is a statement (equation) that two ratios (fractions) are equal. The following property is a convenient way to solve proportions:is equivalent toprovided b ≠ 0 and d ≠ 0.
4ExampleOn 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.
5ExampleA 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 fth8 ft32 ftThe tree is 24 feet tall.
6ExampleA 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. SolutionTotal # taggedTagged in sampleTotal # in sampleTotal # of birds in forestTo the nearest hundred, there are approximately 2900 birds in the forest.
9ExampleLet 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.
10ExampleThe 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.HoursPay6$13811$25315$34523$52931$713
11Example (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 $437.00y = 437
13ExampleLet 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.
14ExampleDetermine 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.x37912y283248As x increases, y decreases. Because xy = 60 for each data point, the equation y = 60/x models the data. This represents an inverse variation.x5101215y64The equation y = 6x models the data. The data represents direct variation.x8111421y486684126
15JOINT VARIATIONLet x, y, and z denote three quantities. Then z varies jointly with x and y if there is a nonzero number k such that
16ExampleThe 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.
17Example (cont)Thus S = 14wt2 models the strength of this type of beam. When w = 4 and t = 3, the beam can supportS = 14 ∙ 4 ∙ 32= 504 pounds