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2.8 Modeling Using Variation Pg. 364 #2-10 (evens), 22-34 (evens)

2.8 Modeling Using Variation Pg. 364 #2-10 (evens), 22-34 (evens). Objectives Solve direct variation problems. Solve inverse variation problems. Solve combined variation problems. Solve joint variation problems. Direct Variation.

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2.8 Modeling Using Variation Pg. 364 #2-10 (evens), 22-34 (evens)

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  1. 2.8 Modeling Using VariationPg. 364 #2-10 (evens), 22-34 (evens) • Objectives • Solve direct variation problems. • Solve inverse variation problems. • Solve combined variation problems. • Solve joint variation problems.

  2. Direct Variation • y varies directly as x (y is directly proportional to x) if y = kx. • k is the constant of variation (also called the constant of proportionality) • As x gets bigger, y gets bigger. • As x gets smaller, y gets smaller. • Example: You are paid $8/hr. Thus, pay is directly related to hours worked: pay=8(hours worked) The more hours worked, the more you get paid. • Example: The smaller the vehicle, the fewer the number of people it can contain. • Example: As a city’s population grows, so do the number of public schools. • Example: When demand for an item increases, the price for the item increases as well.

  3. Process for finding the unknown value with direct variation. • Write an equation of the form y=kx • Use known values to calculate k • Substitute the value for k into y=kx • Answer the problem’s question Direct Variation Question: The number of gallons of water, W, used when taking a shower varies directly with the time, t, in minutes, in the shower. A shower lasting 5 minutes uses 30 gallons of water. How much water is used in a shower lasting 11 minutes?

  4. Direct variation COULD involve an nth power of x (no longer linear) y is directly proportional to the nth power of x. Direct Variation with Powers Question: The distance required to stop a car varies directly with the square of its speed. If 200 feet are required to stop a car traveling 60 miles per hour, how many feet are required to stop a car traveling 100 miles per hour?

  5. Inverse Variation • y varies inversely as x • y is inversely proportional to x • k is the constant of variation • As x gets bigger, y gets smaller • As x gets smaller, y gets bigger • Example: The more pages you read in your novel, the fewer pages you have left to read. • Example: The longer a candle has been burning, the lower the height of the candle stick. • Example: The more furniture you bring into your house, the less open space you have. Inverse Variation Question: The length of a violin string varies inversely with the frequency of its vibrations. A violin string 8 inches long vibrates at a frequency of 640 cycles per second. What is the frequency of a 10 inch string?

  6. Combined Variation • y is impacted by TWO variables in TWO different ways. One variable (x) causes y to get bigger, while the other variable (z) causes it to become smaller. • As x gets bigger, y gets bigger • As z gets bigger, y gets smaller. • k must take into account both influences Combined Variation Question: The number of minutes needed to solve a set of variation problems varies directly with the number of problems and inversely as the number of people working to solve the problems. It takes 4 people 32 minutes to solve 16 problems. How many minutes will it take 8 people to solve 24 problems?

  7. Joint Variation Joint variation is a variation in which a variable varies directly with the product of two or more other variables. Joint Variation Question: The volume of a cone, V, varies jointly with its height, h, and the square of its radius, r. A cone with a radius measuring 6 feet and a height measuring 10 feet has a volume of 120π cubic feet. Find the volume of a cone having a radius of 12 feet and a height of 2 feet.

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