Variation Problems

Variation problems describe how one quantity changes in relation to another. These are common in science, engineering, and economics, where quantities are linked proportionally or inversely.

The constant of variation, usually denoted as $k$, defines the strength or rate of the relationship.

Types of Variation:

Direct Variation: As one variable increases, the other increases proportionally.

$$ y = kx $$

Inverse Variation: As one variable increases, the other decreases proportionally.

$$ y = \frac{k}{x} $$

Joint Variation: One variable varies directly with the product of two or more variables.

$$ y = kxz $$

Combined Variation: A combination of direct and inverse variation.

$$ y = \frac{kz}{x} $$

Variation equations allow us to model relationships like force and acceleration, pressure and volume, population and resources, or resistance and current.

Problem 1:

The intensity of light (in foot-candle) varies inversely as the square of x, the distance in feet from the light source. The intensity of light 2 ft from the source is 80 ft-candles. How far away is the source if intensity of light is 5 ft-candles?

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Problem 2:

The weight of a body varies inversely as the square of its distance from the center of the earth. If the radius of the earth is 4000 miles, how much would a 200-pound man weigh 1000 miles above the surface of the earth?

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Problem 3:

Given that y varies inversely as the square of the difference of w and x, and that y=6 when w=3 and x=1, find the equation for y.

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Problem 4:

The weight of a synthetic ball varies directly with the cube of its radius. A ball with a radius of 2 inches weighs 1.20 pounds. Find the weight of a ball of the same material with a 3-inch radius.

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Problem 5:

The maximum weight that a rectangular beam can support varies jointly as its width and the square of its height and inversely as its length. If a beam ½ foot wide, 1/3 foot high, and 10 feet long can support 12 tons, find how much a similar beam can support if the beam is 2/3 foot wide, ½ foot high, and 16 feet long.

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