1. Section 8.9
Variation

2. Objectives Solve problems involving direct variation
Solve problems involving inverse variation
Solve problems involving joint variation
Solve problems involving combined variation

3. Objective 1: Solve Problems Involving Direct Variation
Direct Variation: The words y varies directly as x or y is directly proportional to x means that y = kx for some nonzero constant k. The constant k is called the constant of variation or the constant of proportionality.
Since the formula for direct variation (y = kx) defines a linear function, its graph is always a line with a y-intercept at the origin. See the graph of y = kx where x ≥ 0 for three positive values of k.

4. Currency Exchange. The currency calculator shown here converts from U
Currency Exchange. The currency calculator shown here converts from U.S. dollars to Russian rubles. When exchanging these currencies, the number of rubles received is
directly proportional to the number of dollars to be exchanged. How many rubles will an
exchange of $1,200 bring?
EXAMPLE 1
Strategy We will use a direct variation model to solve this problem.
Why The words the number of rubles received is directly proportional to the number of dollars to be exchanged indicate that this type of model should be used.

5. Currency Exchange. The currency calculator shown here converts from U
Currency Exchange. The currency calculator shown here converts from U.S. dollars to Russian rubles. When exchanging these currencies, the number of rubles received is directly proportional to the number of dollars to be exchanged. How many rubles will an exchange of $1,200 bring?
EXAMPLE 1
Solution
Step 1: The verbal model can be represented by the equation
Where r is the number of rubles, k is the constant of variation, and d is the number of dollars.
Step 2: From the illustration, we see that an exchange of $500 brings 14,900 rubles. To find k, we substitute 500 for d and 14,900 for r, and then we solve for k .

6. Currency Exchange. The currency calculator shown here converts from U
Currency Exchange. The currency calculator shown here converts from U.S. dollars to Russian rubles. When exchanging these currencies, the number of rubles received is directly proportional to the number of dollars to be exchanged. How many rubles will an exchange of $1,200 bring?
EXAMPLE 1
Solution
Step 3: Now we substitute the value of k, 29.8, into the equation r = kd, to get
Step 4: To find how many rubles an exchange of $1,200 will bring, we substitute 1,200 for d in the direct variation model, and then we evaluate the right side.
An exchange of $1,200 will bring 35,760 rubles.

7. Objective 2: Solve Problems Involving Inverse Variation
Inverse Variation: The words y varies inversely as x or y is inversely proportional to x mean that for some nonzero constant k. The constant k is called the constant of variation.
The formula for inverse variation, , defines a rational function whose graph will have the x- and y-axes as asymptotes. See the graph of where x > 0 for three positive values of k.

8. Photography. The intensity of light received from a light source varies inversely as
the square of the distance from the light source. If a photographer, 16 feet away from his subject, has a light meter reading of 4 foot-candles of luminance, what will the meter read if the photographer moves in for a close-up 4 feet away from the subject?
EXAMPLE 2
Strategy We will use the inverse variation model of the form , where represents the intensity and d2 represents the square of the distance from the light source.
Why The words intensity varies inversely as the square of the distance indicate that this type of model should be used.

9. Photography. The intensity of light received from a light source varies inversely as
the square of the distance from the light source. If a photographer, 16 feet away from his subject, has a light meter reading of 4 foot-candles of luminance, what will the meter read if the photographer moves in for a close-up 4 feet away from the subject?
EXAMPLE 2
Solution
To find the intensity when the photographer is 4 feet away from the subject, we substitute 4 for d and 1,024 for k and simplify.
The intensity at 4 feet is 64 foot-candles.

10. Objective 3: Solve Problems Involving Joint Variation
Joint Variation: If one variable varies directly as the product of two or more variables, the relationship is called joint variation. If y varies jointly with x and z, then y = kxz. The nonzero constant k is called the constant of variation.
There are times when one variable varies as the product of several variables. For example, the area of a triangle varies directly with the product of its base and height:
Such variation is an example of joint variation.

11. Force of the Wind. The force of the wind on a billboard varies jointly as the area of
the billboard and the square of the wind velocity. When the wind is blowing at 20 mph, the force on a billboard 30 feet wide and 18 feet high is 972 pounds. Find the force on a
billboard having an area of 300 square feet caused by a 40-mph wind.
EXAMPLE 3
Strategy We will use the joint variation model ƒ = kAv2, where ƒ represents the force of the wind, A represents the area of the billboard, and v2 represents the square of the velocity of the wind.
Why The words the force of the wind on a billboard varies jointly as the area of the billboard and the square of the wind velocity indicate that this type of model should be used.

12. Force of the Wind. The force of the wind on a billboard varies jointly as the area of
the billboard and the square of the wind velocity. When the wind is blowing at 20 mph, the force on a billboard 30 feet wide and 18 feet high is 972 pounds. Find the force on a
billboard having an area of 300 square feet caused by a 40-mph wind.
EXAMPLE 3
Solution
Since the billboard is 30 feet wide and 18 feet high, it has an area of 30  18 = 540 square feet. We can find k by substituting 972 for ƒ, 540 for A, and 20 for v.

13. Force of the Wind. The force of the wind on a billboard varies jointly as the area of
the billboard and the square of the wind velocity. When the wind is blowing at 20 mph, the force on a billboard 30 feet wide and 18 feet high is 972 pounds. Find the force on a
billboard having an area of 300 square feet caused by a 40-mph wind.
EXAMPLE 3
Solution
To find the force exerted on a 300-square-foot billboard by a 40-mph wind, we use the formula ƒ = Av2 and substitute 300 for A and 40 for v.
The 40-mph wind exerts a force of 2,160 pounds on the billboard.

14. Objective 4: Solve Problems Involving Combined Variation
Many applied problems involve a combination of direct and inverse variation. Such variation is called combined variation.

15. Highway Construction. The time it takes to build a highway varies directly as the length of the road, and inversely as the number of workers. If it takes 100 workers 4 weeks to build 2 miles of highway, how long will it take 80 workers to build 10 miles of highway?
EXAMPLE 4
Strategy We will use the combined variation model , where t represents the time in days, l represents the length of road built in miles, and w represents the number of workers.
Why The words the time it takes to build a highway varies directly as the length of the road, and inversely with the number of workers indicate that this type of model should be used.

16. Highway Construction. The time it takes to build a highway varies directly as the length of the road, and inversely as the number of workers. If it takes 100 workers 4 weeks to build 2 miles of highway, how long will it take 80 workers to build 10 miles of highway?
EXAMPLE 4
Solution
The relationship between these variables can be expressed by the equation

17. Highway Construction. The time it takes to build a highway varies directly as the length of the road, and inversely as the number of workers. If it takes 100 workers 4 weeks to build 2 miles of highway, how long will it take 80 workers to build 10 miles of highway?
EXAMPLE 4
Solution
We now substitute 80 for w, 10 for l, and 200 for k in the equation and simplify:
It will take 25 weeks for 80 workers to build 10 miles of highway.