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6.7 Variation an Problem Solving

Direct Variation y varies directly as x, or y is directly proportional to x, if there is a nonzero constant k such that y = kx. The number k is called the constant of variation or the constant of proportionality. The family of equations of the form y = kx are referred to as direct variation equations.

Suppose that y varies directly as z. If y is 5 when x is 30, find the constant of variation and the direct variation equation. y = kx 5 = k(30) k = 1/6 y = 1616 xSo the direct variation equation is Example

If y varies directly as x, and y = 48 when x = 6, then find y when x = 15. y = kx 48 = k·6 8 = k So the equation is y = 8x. y = 8·15 y = 120 Example

Hooke’s law states that the distance a spring stretches is directly proportional to the weight attached to the spring. If a 40-pound weight attached to the spring stretches the spring 5 inches, find the distance that a 65-pound weight attached to the spring stretches the spring. Continued Example

Substitute the given values. Example (cont) Simplify. Translate the problem into an equation. The equation is: how far the spring stretches

y varies inversely as x, or y is inversely proportional to x, if there is a nonzero constant k such that The number k is called the constant of variation or the constant of proportionality. Inverse Variation

Suppose that y varies inversely as x. If y = 63 when x = 3, find the constant of variation k and the inverse variation equation. k = 63·3 k = 189 Example So the inverse variation equation is y = 189 x.

The maximum weight that a circular column can hold is inversely proportional to the square of its height. If an 8-foot column can hold 2 tons, find how much weight a 10-foot column can hold. Continued Example

So our equation is We substitute our given value for the height of the column into the equation. Example (cont)

If the ratio of a variable y to the product of two or more variables is constant, then y varies jointly as, or is jointly proportional to, the other variables. If y = kxz then the number k is the constant of variation or the constant of proportionality. Joint Variation

The lateral surface area of a cylinder varies jointly as its radius and height. Express surface area S in terms of radius r and height h. S = krh Example

Some examples of variation involve combinations of direct, inverse, and joint variation. We call these variations combined variation. Combined Variation

At sea, the distance to the horizon is directly proportional to the square root of the elevation of the observer. If a person who is 36 feet above water can see 7.4 miles, find how far a person 64 feet above the water can see. Round your answer to two decimal places. Example Continued

So our equation is We substitute our given value for the elevation into the equation. Example (cont)