Lesson 37 Combined Variation Combined Variation: - a combination of direct and indirect variation, or joint and indirect variation o when a quantity varies directly (or jointly) with one or more variables and inversely with one or more variables 푘푥푧 - described by formulas such as 푦 = , where 푦 varies directly with 푥 푤 and 푧 and inversely with 푤 o depending on how the variables in the numerator and/or denominator change (increasing or decreasing), the dependent variable could increase, decrease, or remain unchanged - Example of combined variation: o Newton’s law of universal gravitation 푘∙푚 ∙푚 . the formula is 퐹 = 1 2, where 퐹 is the gravitational 푑2 force between two objects, 푚1 is the mass of one object, 푚2 is the mass of another object, 푑 is the distance between the two objects, and 푘 is the constant of variation . for this formula, the constant of variation 푘 is the gravitational constant (0.000000000066743…) Example 1: Express the following statement as a formula that involves the given variables and a constant of proportionality 푘, and then determine the value of 푘 from the given conditions. 푟 varies directly as 푠 and indirectly as 푡. If 푠 = 2 and 푡 = 4, then 푟 = 7. 1 Lesson 37 Combined Variation Example 2: Express the following statement as a formula that involves the given variables and a constant of proportionality 푘, and then determine the value of 푘 from the given conditions. 푦 is jointly proportional to the square root of 푥 and the cube of 푧, and inversely proportional to the 푤5. If 푥 = 9, 푧 = 5, and 푤 = 2, then 푦 = 17. Example 3: The centrifugal force 퐹 of a body moving in a circle varies jointly with the radius 푟 of the circular path and the body’s mass 푚, and inversely with the square of the time 푡 it takes to move about one full circle. a. Express the statement above as a formula. b. A 6-gram body moving in a circle with radius 100 centimeters at a rate of 1 revolution every 2 seconds has a centrifugal force of 6,000 dynes. Use this information to determine the value of 푘. c. Find the centrifugal force of an 18-gram body moving in a circle with radius 100 centimeters at a rate of 1 revolution every 3 seconds? 2 Lesson 37 Combined Variation Example 4: In baseball, a pitcher’s earned-run average 퐴 varies directly as the number of earned runs 푅 allowed and inversely as the number of innings pitched 퐼. a. Express the previous statement as a formula. b. If a pitcher has an earned-run average of 3.6 after pitching 95 innings and allowing 38 earned-runs, what is the value of 푘? 푘(38) 3.6 = 95 95(3.6) = 38푘 95(3.6) = 푘 38 푘 = 9 c. What is the earned-run average of a pitcher who gave up 69 earned runs in 308 innings? Round to the hundredths place. 9(69) 퐴 = 308 621 퐴 = 308 퐴 ≈ 2.02 Answers to Examples: 푘푠 푘√푥푧3 544 1. 푟 = ; 푘 = 14 ; 2. 푦 = ; 푘 = ; 푡 푤5 375 푘∙푟∙푚 3a. 퐶 = ; 3b. 푘 = 40 ; 3c. 퐶 = 8,000 ; 푡2 푘∙푅 4a. 퐴 = ; 4b. 푘 = 9 ; 4c. 퐴 = 2.02 ; 퐼 3 .