<p> Variation and Problem Solving – Section 8.4</p><p>Direct Variation: _____ varies ______as _____, or y is directly proportional to x, if there is a nonzero constant k such that</p><p>The number k is called the ______of ______.</p><p>Example 1: Suppose that y varies directly as x. If y is 5 when x is 30, find the constant of variation and the direct variation equation.</p><p>Example 2: If m varies directly as x2 and m is 6 when x is 36, find the constant of variation. Then write the direct variation equation. </p><p>Your Turn: Suppose that y varies directly as x. If y is 20 when x is 4, find the constant of variation and the direct variation equation.</p><p>Inverse Variation: _____ varies ______as _____, or y is inversely proportional to x, if there is a nonzero constant k such that</p><p>**k is still known as the constant of variation.</p><p>Example 3: Suppose that u varies inversely with w. If u is 3 when w is 5, find the constant of variation and the inverse variation equation.</p><p>Example 4: The variables of x and y vary inversely. What is the inverse variation that relates the variables if x = 4 and y = -3? Your Turn: Suppose that a varies inversely with b. If a is 8 when b is 2, find the constant of variation and the inverse variation equation.</p><p>Joint Variation: If the ratio of a variable y to the product of two or more variables in constant, then y varies jointly as, or is jointly proportional to, the other variables. If</p><p>Then the number k is the constant variation. </p><p>Example 5: Suppose x varies jointly with y and z. If x is 36 when y is 3 and z if 2, find the constant of variation and the joint variation equation.</p><p>Example 6: Suppose a varies jointly with c2 and m, and a = 12 when c = 2 and m = 9. Find m when a = 15 and c = 3. </p><p>Your Turn: Suppose a varies jointly with b and c. If a is 45 when b is 5 and c if 3, find the constant of variation and the joint variation equation.</p><p>Combined Variation: Variation that involves combinations of direct, inverse, and joint variation. </p><p>Example 7: Suppose y is directly proportional to x2 and inversely proportional to z. If y is 24 when x is 4 and z is 2, find the constant variation and the variation equation. Example 8: Suppose b is directly proportional to a and inversely proportional to c. If b is 10 when c is 2 and a is 4, find the constant variation and the variation equation. </p><p>Application: 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.</p><p>Application: Boyle’s law says that if the temperature stays the same, the pressure P of a gas is inversely proportional to the volume V. If a cylinder in a steam engine has a pressure of 960 kilopascals when the volume is 1.4 cubic meters, find the pressure when the volume increases to 2.5 cubic meters.</p><p>Application: The power P in watts of an electrical circuit varies jointly as the resistance R and the square of the current I. For a 600-watt microwave oven that draws a current of 5.0 amperes, the resistance is 24 ohms. What is the resistance of a 200-watt refrigerator that draws a current of 2.5 amperes? </p>