DIRECT, INVERSE, DIRECT, INVERSE, AND JOINT VARIATION AND JOINT VARIATION

Unit 3Unit 3English CasbarroEnglish Casbarro

If y varies directly as x and y = 14, when x = 3.5, find k and write and graph the variation.

Notice that the direct variation graph always intersectsthe coordinate plane at the origin. Try this: If y varies directly as x and y = 6.5, when x = 13, find k and graph the variation.

When you have to find values for a variation, you can set up a proportion equation.Since the constant of proportionality never changes, you can solve the directvariation equation for k. That k will be the same no matter how the x and y values change.

Example 2

The circumference of a circle, C, varies directly as the radius,r and C=7π ft.when r = 3.5 ft. Find r when C=4.5π ft.

The perimeter, P, varies directly as the side length, s and P = 18 in as s = 1.5 in.Find s when P = 75 in.

Example 3

Think of factors of a number like 24. As one side goes up, the other side goes down, but they still equal 24. This is called inverse variation. As x increases, y decreases,or as x decreases, y increases. The equation for inverse variation is

xy = kExample 4 y varies inversely as x and y = 3 when x = 8.

Find and graph the inverse variation function.

Because you cannot divide by 0, the function is undefined at x = 0 (there is an asymptote).

When you have to find values for a variation, you can set up an equation.Since the constant of proportionality never changes, k will be the same no matter

how the x and y values change.

So, x1y1=x2y2

Example 5

The time, t, it takes volunteers to build a house varies inversely with the number of volunteers, v. If 20 volunteers can build a house in 62.5 hours, how many volunteers would be needed to build a house in 50 hours?

Example 6 How long would it take 15 volunteers to build a house?

Direct/divide

A joint variation is a relationship between 3 variables, and the equation looks like this:

y=kxzExample 7 The area, A, of a triangle varies jointly as the base, b, and the height, h

and A = 12 m2 when b = 6 m and h = 4 m. Find b when A = 36 m2 andh = 8m.

Example 8 The lateral surface area, L, of a cone varies jointly as the base

radius, r, and the slant height, l and L = 63π m2 when r = 3.5 m and

l = 18 m. Find r to the nearest tenth when L = 8π m2 and l = 5 m.

Combined variation is a combination of both direct and inverse variation. You would Write all inverse variation on the same side of the equals sign, and the direct variation on the opposite side of the equals sign.

So, it looks like this: (inverse) y= k (direct)

Example 9

The volume of a gas, V, varies inversely as pressure, P, and directly as temperature, T.

a certain gas has a volume of 10 l (liters), a temperature of 300 K (kelvins) and a pressure

of 1.5 atm(atmospheres). If the gas is compressed to 7.5 l and is heated to 350 K, whatwill the new pressure be?

Example 10

If the gas is heated to 400 K and has s pressure of 1 atm., what is the pressure?

Example 11

The power, P, that must be delivered to a car engine varies directly as the distance, d, that the car moves and inversely as the time, t, required to move that distance. To move the car 500 m in 50 s, the engine must deliver 147 kW of power. How many kW must the engine deliver to move the car 700 m in 30 s?

Turn in the following problems

1. Which is an example of inverse variation? a. The total cost as a function of the number of items purchased

b. The distance travelled as a function of speed. c. The area of a swimming pool as a function of its radius. d. The number of posts in a 20-ft. fence as a function of distance between posts.2. Which equation is best represented by the statement: y varies directly as the square root of x.

3. Which statement is best represented by the graph? a. y varies directly as x2. b. y varies inversely as x. c. y varies directly as x. d. x varies inversely as y.

4, In an auto race, a car with a speed of 200 mi/h takes an average of 31.5 s to complete one lap of the track. a. Write an inverse variation function that gives the average speed, s, of the car in miles per hour as a function of time, t, to complete one lap. b. How many seconds does it take the car to complete one lap at 210 mph?