1. 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.

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. 1.Write an equation of the form y=kx 2.Use known values to calculate k 3.Substitute the value for k into y=kx 4.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.