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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