171S3.8.notebook
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MAT 171
3.8 Variation: Function Models in Action
A. Toolbox Functions and Direct Variation
Direct Variation y varies directly with x, or y is directly proportional to x, if there is a nonzero constant k such that y = kx. k is called the constant of variation.
Solving Applications of Variation 1. Write the information given as an equation, using k as the constant multiple. 2. Substitute the first relationship ( pair of values) given and solve for k. 3. Substitute this value for k in the original equation to obtain the variation equation. 4. Use the variation equation to complete the application.
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B. Inverse Variation
Inverse Variation y varies inversely with x, or y is inversely proportional to x, if there is a nonzero constant k such that y=k(1/x)
k is called the constant of variation.
C. Joint or Combined Variations
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171S3.8.notebook
March 23, 2010
321/8. Write the variation equation for each statement: cost varies directly with the quantity purchased
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321/10. Write the variation equation for each statement: length of a spring varies directly with attached weight
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171S3.8.notebook
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321/11. Find the constant of variation and write the variation equation. Then use the equation to complete the table: y varies directly with x; y = 0.6 when x = 24.
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322/12. Find the constant of variation and write the variation equation. Then use the equation to complete the table: w varies directly with v; w = 1/3 when v = 5.
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322/14. The thickness of a paperback book varies directly as the number of pages. A book 3.2 cm thick has 750 pages. Write the variation equation and approximate the thickness of Roget’s 21st Century Thesaurus ( paperback— 2nd edition), which has 957 pages.
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322/16. The height of a projected image varies directly as the distance of the projector from the screen. At a distance of 48 in., the image on the screen is 16 in. high. ( a) Find the constant of variation and write the variation equation, ( b) graph the variation equation, ( c) use the graph to estimate the height of the image if the projector is placed at a distance of 5 ft 3 in., and ( d) use the equation to check this estimate. Was it close?
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171S3.8.notebook
March 23, 2010
322/18. Write the variation equation for each statement. Potential energy in a spring varies directly with the square of the distance the spring is compressed.
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322/20. Write the variation equation for each statement. Manufacturing cost varies directly as the square of the number of items made.
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171S3.8.notebook
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322/24. Find the constant of variation and write the variation equation. Then use the equation to complete the table or solve the application. The area of an equilateral triangle varies directly as the square of one side. A triangle with sides of 50 yd has an area of 1082.5 mi2. Find the area in mi2 of the region bounded by straight lines connecting the cities of Cincinnati, Ohio, Washington, D. C., and Columbia, South Carolina, which are each approximately 400 mi apart.
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322/26. Find the constant of variation and write the variation equation. Then use the equation to complete the table or solve the application. When a child blows small soap bubbles, they come out in the form of a sphere because the surface tension in the soap seeks to minimize the surface area. The surface area of any sphere varies directly with the square of its radius. A soap bubble with a in. radius has a surface area of approximately 7.07 in2. ( a) Find the constant of variation and write the variation equation, Mar 15­10:52 AM
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322/26. ( b) graph the variation equation, ( c) use the graph to estimate the radius of a seventeenth­ century cannonball that has a surface area of 113.1 in2 and ( d) use the equation to check this estimate. Was it close? ( e) According to the equation, what is the surface area of an orange with a radius of 1 1/2 in.?
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323/32. Find the constant of variation and write the variation equation. Then use the equation to complete the table or solve the application. A varies inversely with B; A = 2450 when B = 0.8
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171S3.8.notebook
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323/34. Find the constant of variation and write the variation equation. Then use the equation to complete the table or solve the application. The demand for a popular new running shoe varies inversely with the cost of the shoes. When the wholesale price is set at $ 45, the manufacturer ships 5500 orders per week to retail outlets. Based on this information, how many orders would be shipped if the wholesale price rose to $ 55?
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323/36. Write the variation equation for each statement. Horsepower varies jointly as the number of cylinders in the engine and the square of the cylinder’s diameter.
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323/42. Find the constant of variation and write the related variation equation. Then use the equation to complete the table or solve the application. J varies directly with P and inversely with the square root of Q, J = 19 when P = 4 and Q = 25.
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323/44. Find the constant of variation and write the related variation equation. Then use the equation to complete the table or solve the application. Safe load: The load that a horizontal beam can support varies jointly as the width of the beam, the square of its height, and inversely as the length of the beam. A beam 4 in. wide and 8 in. tall can safely support a load of 1 ton when the beam has a length of 12 ft. How much could a similar beam 10 in. tall safely support?
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324/52. Supply and demand: A chain of hardware stores finds that the demand for a special power tool varies inversely with the advertised price of the tool. If the price is advertised at $ 85, there is a monthly demand for 10,000 units at all participating stores. Find the projected demand if the price were lowered to $ 70.83.
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324/53. Cost of copper tubing: The cost of copper tubing varies jointly with the length and the diameter of the tube. If a 36­ ft spool of 1/4 ­in­ diameter tubing costs $ 76.50, how much does a 24­ ft spool of 3/8 ­in.­diameter tubing cost?
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