Certain situations exist where:  If one quantity increases, the other quantity also increases.  If one quantity increases, the other quantity decreases.

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Presentation transcript:

Certain situations exist where:  If one quantity increases, the other quantity also increases.  If one quantity increases, the other quantity decreases. This kind of modeling is called variation. 2.5 Variation

Direct Variation Definition: When one quantity increases, the other increases. When one quantity decreases, the other decreases. y varies directly as x. y is directly proportional to x. k is the constant of proportionality

Indirect Variation Definition: When one quantity increases, the other decreases. When one quantity decreases, the other increases. y varies indirectly (or inversely) as x. y is indirectly proportional to x. k is the constant of proportionality

Joint Variation Definition: When one quantity varies according to two other quantities. z varies directly as x and y. z is jointly proportional to x and y. z varies directly as x and indirectly as y.

Example 1 Express the statement as an equation. Use the given information to find the constant of proportionality. y is directly proportional to x. When x = 3, y = 39.

Example 2 Express the statement as an equation. Use the given information to find the constant of proportionality. M is inversely proportional to the square of t. When t = -4, M = 5.

Example 3 C is proportional to cube root of s and inversely proportional to t. When t = 7 and s = 27, C = 66. Express the statement as an equation. Use the given information to find the constant of proportionality.

Example 4 The pressure P of a sample of gas is directly proportional to the temperature T and inversely proportional to the volume V. Write an equation that expresses this variation. Find the constant of proportionality if 125 L of gas exerts a pressure of 12.7 kPa at a temperature of 250 K. Find the pressure if volume is increased to 150 L and temperature is decreased to 200 K. (Round the answer to the nearest tenth.)

Example 5 The force F needed to keep a car from skidding is jointly proportional to the weight w of the car and the square of the speed s, and is inversely proportional to the radius of the curve r. Write an equation that expresses this variation. A car weighing 3,600 lb travels around a curve at 60 mph. The next car to round the curve weighs 2,500 lb and requires the same force as the first. How fast was the second car traveling?

Example 5(continued) First Car(3600lb, 60mph)Second Car(2500lb, ?mph)