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1 PRECALCULUS I Dr. Claude S. Moore Danville Community College Mathematical Modeling Direct, inverse, joint variations; Least squares regression

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1. y varies directly as x. 2. y is directly proportional to x. 3. y = kx for some nonzero constant m. NOTE: k is the constant of variation or the constant of proportionality. Example: If y = 3 when x = 2, find k. y = kx yields 3 = m(2) or m = 1.5. Thus, y = 1.5x. Direct Variation Statements

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1. y varies directly as the nth power of x. 2. y is directly proportional to the nth power of x. 3. y = kx n for some nonzero constant k. NOTE: k is the constant of variation or constant of proportionality. Direct Variation as nth Power

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1. y varies inversely as x. 2. y is inversely proportional to x. 3. y = k / x for some nonzero constant k. NOTE: k is the constant of variation or the constant of proportionality. Example: If y = 3 when x = 2, find k. y = k / x yields 3 = k / 2 or k = 6. Thus, y = 6 / x. Inverse Variation Statements

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1. z varies jointly as x and y. 2. z is jointly proportional to x and y. 3. z = kxy for some nonzero constant k. NOTE: k is the constant of variation. Example: If z = 15 when x = 2 and y = 3, find k. z = kxy yields 15 = k(2)(3) or k = 15/6 = 2.5. Thus, z = 2.5xy. Joint Variation Statements

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B ICYCLING A bicyclist tips the bicycle when making turn. The angle B of the bicycle from the vertical direction is called the banking angle. U SING D IRECT AND I NVERSE V ARIATION IN R EAL L IFE Writing and Using a Model banking angle, B

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Writing and Using a Model From the graph, you can see that B = 32 ° when r = 3.5 feet. B =B = k r 32 = k = k S OLUTION Turning Radius Banking angle (degrees) Write direct variation model. Substitute 32 for B and 3.5 for r. Solve for k. Find an inverse variation model that relates B and r. The model is B =, where B is in degrees and r is in feet. 112 r

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Writing and Using a Model S OLUTION Use the model to find the banking angle for a turning radius of 5 feet. Substitute 5 for r in the model you just found. B =B = = 22.4 Turning Radius Banking angle (degrees ) When the turning radius is 5 feet, the banking angle is about 22 °.

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