Section 6 5 Variation Copyright 2013 2010 2007

What You Will Learn Direct Variation Inverse Variation Joint Variation 6. 5 -2 Copyright

Variation A variation equation is an equation that relates one variable to one or

Direct Variation In direct variation, the values of the two related variables increase or

Direct Variation If a variable y varies directly with a variable x, then y

Example 1: Direct Variation in Electronics The resistance of a wire, R, varies directly

Example 4: Using the Constant of Proportionality The area, a, of a picture projected

Example 4: Using the Constant of Proportionality Solution First determine k. Use k =

Example 4: Using the Constant of Proportionality Solution Thus, the area of a projected

Inverse Variation When two quantities vary inversely, as one quantity increases, the other quantity

Inverse Variation If a variable y varies inversely with a variable, x, then where

Example 5: Inverse Variance in Illuminance The illuminance, I, of a light source varies

Example 5: Inverse Variance in Illuminance Solution To determine k, use the general form

Joint Variation One quantity may vary directly as the product of two or more

Joint Variation The general form of a joint variation, where y, varies directly as

Example 7: Joint Variation The area, A, of a triangle varies jointly as its

Example 7: Joint Variation Solution 6. 5 -17 Copyright 2013, 2010, 2007, Pearson, Education,

Example 7: Joint Variation Solution 6. 5 -18 Copyright 2013, 2010, 2007, Pearson, Education,

Combined Variation Often in real-life situations, one variable varies as a combination of variables.

Example 8: Combined Variation in Engineering The load, L, that a horizontal beam can

Example 8: Combined Variation in Engineering Solution 6. 5 -21 Copyright 2013, 2010, 2007,

Example 9: Hot Dog Price, Combined Variation The owners of Henrietta Hots find their

Example 9: Hot Dog Price, Combined Variation a) Write a variation expressing S in

Example 9: Hot Dog Price, Combined Variation Solution The equation for weekly sales of

Example 9: Hot Dog Price, Combined Variation Solution Henrietta Hots can expect to sell

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Variation A variation equation is an equation that relates one variable to one or more other variables through the operations of multiplication or division or both operations. 6. 5 -3 Copyright 2013, 2010, 2007, Pearson, Education, Inc.

Direct Variation In direct variation, the values of the two related variables increase or decrease together. As one increases so does the other As one decreases so does the other 6. 5 -4 Copyright 2013, 2010, 2007, Pearson, Education, Inc.

Direct Variation If a variable y varies directly with a variable x, then y = kx where k is the constant of proportionality (or the variation constant). 6. 5 -5 Copyright 2013, 2010, 2007, Pearson, Education, Inc.

Example 1: Direct Variation in Electronics The resistance of a wire, R, varies directly as its length, L. Write an equation for the resistance of a wire, R, if the constant of proportionality is 0. 008. Solution R = k. L R = 0. 008 L 6. 5 -6 Copyright 2013, 2010, 2007, Pearson, Education, Inc.

Example 4: Using the Constant of Proportionality The area, a, of a picture projected on a movie screen varies directly as the square of the distance, d, from the projector to the screen. If a projector at a distance of 25 feet projects a picture with an area of 100 square feet, what is the area of the projected picture when the projector is at a distance of 40 feet? 6. 5 -7 Copyright 2013, 2010, 2007, Pearson, Education, Inc.

Example 4: Using the Constant of Proportionality Solution Thus, the area of a projected picture is 256 ft 2 when the projector is at a distance of 40 ft. 6. 5 -9 Copyright 2013, 2010, 2007, Pearson, Education, Inc.

Inverse Variation When two quantities vary inversely, as one quantity increases, the other quantity decreases, and vice versa. Also referred to as inversely proportional. 6. 5 -10 Copyright 2013, 2010, 2007, Pearson, Education, Inc.

Example 5: Inverse Variance in Illuminance The illuminance, I, of a light source varies inversely as the square of the distance, d, from the source. If the illuminance is 80 units at a distance of 5 meters, determine the equation that expresses the relationship between the illuminance and the distance. 6. 5 -12 Copyright 2013, 2010, 2007, Pearson, Education, Inc.

Joint Variation The general form of a joint variation, where y, varies directly as x and z, is y = kxz where k is the constant of proportionality. 6. 5 -15 Copyright 2013, 2010, 2007, Pearson, Education, Inc.

Example 7: Joint Variation The area, A, of a triangle varies jointly as its base, b, and height, h. If the area of a triangle is 48 in. 2 when its base is 12 in. and its height is 8 in. , find the area of a triangle whose base is 15 in. and whose height is 20 in. 6. 5 -16 Copyright 2013, 2010, 2007, Pearson, Education, Inc.

Example 8: Combined Variation in Engineering The load, L, that a horizontal beam can safely support varies jointly as the width, w, and the square of the depth, d, and inversely as the length, l. Express L in terms of w, d, l, and the constant of proportionality, k. 6. 5 -20 Copyright 2013, 2010, 2007, Pearson, Education, Inc.

Example 9: Hot Dog Price, Combined Variation The owners of Henrietta Hots find their weekly sales of hot dogs, S, vary directly with their advertising budget, A, and inversely with their hot dog price, P. When their advertising budget is $600 and the price of a hot dog is $1. 50, they sell 5600 hot dogs a week. 6. 5 -22 Copyright 2013, 2010, 2007, Pearson, Education, Inc.

Example 9: Hot Dog Price, Combined Variation a) Write a variation expressing S in terms of A and P. Include the value of the constant of proportionality. b) Find the expected sales if the advertising budget is $800 and the hot dog price is $1. 75. 6. 5 -23 Copyright 2013, 2010, 2007, Pearson, Education, Inc.

Example 9: Hot Dog Price, Combined Variation Solution Henrietta Hots can expect to sell 6400 hot dogs a week if the advertising budget is $800 and the hot dog price is $1. 75. 6. 5 -25 Copyright 2013, 2010, 2007, Pearson, Education, Inc.