Presentation on theme: "Modeling Using Variation"— Presentation transcript:
1 Modeling Using Variation Section 3.7Modeling Using VariationObjectives:Solve direct variation problems.Solve inverse variation problems.Solve combined variation problems.
2 Example of Direct Variation Because light travels faster than sound, during a thunderstorm we see lightning before we hear thunder. The formula d = 1080t describes the distance, in feet, of the storm’s center if it takes t seconds to hear thunder after seeing lightning. Thus if t = 1, d = 1080; if t = 2, d = 2160; and if t = 3, d = This distance is said to vary directly.
3 Direct VariationIf a situation is described by an equation in the formy = kxwhere k is a constant, we say thaty varies directly as x. The number k is called the constant of variation.As x increases, y increases. As x decreases, y decreases.
4 Writing a Direct Variation Equation If v varies directly as r, then:Solution:v = kr
5 Solving Variation Problems Write an equation that describes the given English statement.Substitute the given pair of values into the equation in step 1 and find the value of k.Substitute the value of k into the equation in step 1.Use the equation from step 3 to answer the problem's question.
6 Text ExampleThe amount of garbage, G, varies directly as the population, P. Allegheny County, Pennsylvania, has a population of 1.3 million and creates 26 million pounds of garbage each week. Find the weekly garbage produced by New York City with a population of 7.3 million.SolutionStep Write an equation We know that y varies directly as x is expressed asy = kx.By changing letters, we can write an equation that describes the following English statement: Garbage production, G, varies directly as the population, P.G = kPStep 2 Use the given values to find k. Allegheny County has a population of 1.3 million and creates 26 million pounds of garbage weekly. Substitute 26 for G and 1.3 for P in the direct variation equation. Then solve for k.26 = 1.3k26/1.3 = k20 = k
7 Text Example cont. Step 3 Substitute the value of k into the equation. G = kP Use the equation from step 1.G = 20P Replace k, the constant of variation, with 20.Step 4 Answer the problem's question New York City has a population of 7.3 million. To find its weekly garbage production, substitute 7.3 for P inG = 20P and solve for G.G = 20P Use the equation from step 3.G = 20(7.3) Substitute 7.3 for P.G = 146The weekly garbage produced by New York City weighs approximately 146 million pounds.
8 Practice #1The pressure, P, of water on an object below the surface varies directly as its distance, D below the surface. If a submarine experiences a pressure of 25 pounds per square inch 60 feet below the surface, how much pressure will it experience 330 feet below the surface.
9 Direct Variation with Powers y varies directly as the nth power of x if there exists some nonzero constant k such thaty = kx n.We also say that y is directly proportional to the nth power of x.
10 Practice #2The distance, s, that a body falls from rest varies directly as the square of the time, t, of the fall. If skydivers fall 64 feet in 2 seconds, how far will they fall in 4.5 seconds?
11 Inverse VariationIf a situation is described by an equation in the formwhere k is a constant, we say that y varies inversely as x. The number k is called the constant of variation.As x increases, y decreases. As x decreases, y increases.
12 Combined Variation Example Describe in words the variation shown by the given equation:Solution: H varies directly as T and inversely as Q