do now a car uses 8 gallons of gasoline to travel 290 miles. how much gasoline will the car use to...

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A car uses 8 gallons of gasoline to travel 290 miles. How much gasoline will the car use to travel 400 miles?

Using Direct Variation Method to solve this word problem:Word Problem:

A car uses 8 gallons of gasoline to travel 290 miles. How much gasoline will the car use to travel 400 miles?

Step 1: Find points in table

X (gas) Y (miles) 8 290 ? 400

Step Two: Find the constant variation and equation:

k = y/x or k = 290/8 or 36.25

y = 36.25 x

Step Three: Use the equation to find the unknown.400 =36.25x 400 =36.25x36.25 36.25or x = 11.03

Recall,

The general equation for DIRECT VARIATION is

k is called the constant of variation.

Is this a direct variation? If yes, give the constant of variation (k) and the

equation.

X Y 4 6 8 12

12 18 18 27

Yes!

k = 6/4 or 3/2

Equation?

y = 3/2 x

X Y 15 5 3 26 1 75 2 150

No!

The k values are different!

Is this a direct variation? If yes, give the constant of variation (k) and the

equation.

If y varies directly as x and y=6 when x=5, then find y when x=15.

Direct Variation Method:

If y varies directly as x and y=6 when x=5, then find y when x=15.

Proportion Method:

If y varies directly as x, and y=24 and x=3 find: Find y when x=2

First find the constant of variation

Write the general equation

Substitute

(b) Find y when x=2

First we find the constant of variation, which was k=8

Now we substitute into y=kx.

You can also solve it using the Proportion Method.

How does the graph y=kx look like?What is the constant of variation?

No Yes

Yes No

Tell if the following graph is a Direct Variation or not.

Word problem 2.

According to Hook’s Law, the force F required to stretch a spring x units beyond its natural length varies directly as x. A force of 30 pounds stretches a certain spring 5 inches. Find how far the spring is stretched by a 50 pound weight.

Set up a proportion

Substitute

Now try this problem.

Use Hook’s Law to find how many pounds of force are needed to stretch a spring 15 inches if it takes 18 pounds to stretch it 13.5 inches.

Answer: 20 pounds

Practice

Worksheet: 1-10

Exit Question1. Which of the following is a

direct variation?

2. Given that y varies directly with x, and y = 28 when x=7, Find x when y = 52.

Do Now

Given that y varies directly with x, and y = 6 when x=-5,

Find y when x = -8.

y varies directly as x, and x=8 when y=9. Find y when x=12.

Answer: 13.5

Tell if the following graph is a Direct Variation or not.

No Yes

No No

Inverse Variation

y varies inversely as x if

such that xy=k or

Just as with direct variation, a proportion can be set up solve problems of indirect variation.

Tell whether each relationship is an inverse variation. Explain.

1 2

Find y when x=15, if y varies inversely as x and x=10 when y=12

Solve by equation:

Solve this problem using either method.

Find x when y=27, if y varies inversely as x and x=9 when y=45.

Answer: 15

Lets apply what we have learned.

The pressure P of a compressed gas is inversely proportional to its volume V according to Boyle’s Law. A pressure of 40 pounds per square inch is created by 600 cubic inches of a certain gas. Find the pressure when the gas is compressed to 200 cubic inches.

A pressure of 20 pounds per inch squared is exerted by 400 inches cubed of a certain gas. Use Boyle’s Law to find the pressure of the gas when it is compressed to a volume of 100 inches cubed.

The pressure P of a compressed gas is inversely proportional to its volume V according to Boyle’s Law

What does the graph of xy=k look like?

This is a graph of a hyperbola.

Notice: That in the graph, as the x values increase the y values decrease.

also

As the x values decrease the y values increase.

Practice

• Textbook p. 127- 128/ 1, 2, 5, 6

• Hw: p. 128/ 7, 9