Learn about direct, inverse, and joint variation in mathematical modeling, with practical examples including revenue from gasoline sales and weight calculations at different altitudes. Discover how variables relate to each other and vary either directly, inversely, or jointly in different scenarios.
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Presentation Transcript
Variation Objectives Construct a Model Using Direct Variation Construct a Model Using Inverse Variation Construct a Model Using Joint or Combined Variation
Variation refers to how one quantity varies in relation to another quantity. Quantities may vary directly, inversely, or jointly. Let x and y denote two quantities. Then y varies directly with x, or y is directly proportional to x, if there is a nonzero number k such that y = kx. The number k is called the constant of proportionality.
For regular unleaded gasoline, the revenue R (in dollars) varies directly with the number of gallons of gasoline sold g. If revenue is $15.00 when the number of gallons of gasoline sold is 12.5, find a formula that relates revenue R to the number of gallons of gasoline g. R = kg 15 = k(12.5) 15 125 . k = = 120 . Thus, R = 1.20g
Let x and y denote two quantities. Then y varies inversely with x, or y is inversely proportional to x, if there is a nonzero number k such that: y = k/x The number k is called the constant of proportionality.
The weight of a body varies inversely with the square of its distance from the center of Earth. Assuming the radius of Earth is 3960 miles, how much would a woman weigh at an altitude of 0.5 miles above the Earth s surface if she weighs 120 pounds on Earth s surface? k d k = = 3960 120 1881792 000 , =1881792 000 2 k = W = 120 39602 2 2 , , , , , W So, d 1881792 000 3960 , , , 1881792 000 39605 , , , W = = = 119 97 . ( ( ) ) 2 2 + 05 . .
When a variable quantity Q is proportional to the product of two or more other variables, we say that Q varies jointly with these quantities. Combined variationis a combination of direct and/or inverse variation.
The maximum safe load for a horizontal rectangular beam varies jointly with the width of the beam and the square of the thickness of the beam and inversely with its length. If a 10-foot beam will support up to 600 pounds when the beam is 3 inches wide and 4 inches thick, what is the maximum safe load of a similar beam 12 feet long, 4 inches wide and 6 inches thick? kwt l 3 4 10 2 2 k = 125 = 600 k = M 2 2 1254 6 wt =125 M = = M 1500 l 12
