1. Modeling Using Variation
Section 3.7
Modeling Using Variation
Objectives:
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 Variation
If a situation is described by an equation in the form
y = kx
where k is a constant, we say that
y 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 Example
The 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.
Solution
Step Write an equation We know that y varies directly as x is expressed as
y = 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 = kP
Step 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 = 1.3k
26/1.3 = k
20 = 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 in
G = 20P and solve for G.
G = 20P Use the equation from step 3.
G = 20(7.3) Substitute 7.3 for P.
G = 146
The weekly garbage produced by New York City weighs approximately 146 million pounds.

8. Practice #1
The 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 that
y = kx n.
We also say that y is directly proportional to the nth power of x.

10. Practice #2
The 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 Variation
If a situation is described by an equation in the form
where 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