Algebra 1A



Algebra 1 Name: _______________________________

Direct Variation (DV) Period: ______ Seat: _________

When there are two quantities that depend on each other in such a way that if you double one of them then the other doubles, they are said to “vary directly.” There are many applications of this, and all can be solved by listing the quantities in 2 x 2 tables, writing a ratio, and then cross multiplying to solve. When the quantities are called x and y, then a graph of their values will always be a line that goes through the origin. The slope of this line will be the same as the common ratio of any y-value to the corresponding x-value. So any equation of the form y = mx will represent a direct variation. The slope, m, is called the “constant of variation,” and some books use k instead of m, apparently from the German word, konstant.

Example 1: Albert’s car takes 24 inch tires. One day Albert got some 26 inch tires at a yard sale and put them on his car. Later he was driving on I-40 near Tucumcari where the speed limit was 75 mph. His speedometer showed he was going exactly 75 mph, but because he had larger tires on the car, this was inaccurate. The police stopped him for speeding. How fast was he really going?

Solution: First we need to decide if this is really a direct variation problem. Imagine what would happen if he could have put on tires that were twice as large. This would mean he would go twice as far for one revolution of his wheels, and speedometers only measure the revolutions per unit of time. So he would actually be going twice as fast as his speedometer shows. Therefore, tire-size and true speed vary directly.

Now set up a table, write a ratio and solve:

Ratio: [pic]

Cross-multiply: [pic]

Solve: [pic] [pic] x = 81.25

So he must have been going 81.25 mph. No wonder he got stopped.

Example 2: For a long time, Sally has wondered how tall a utility pole in front of her house is. One morning she had an epiphany and ran out front with a yardstick, which she held vertically on the ground. It was a bright, sunny morning, and the shadow cast by the yardstick was 13 inches long. Then she measured the shadow of the pole and found it was 18 feet long. From this information she figured out how tall the pole was. How tall was it?

Solution: A yard-stick is 36 inches long. Set up a

table, write a ratio, cross-multiply and solve:

shadow height

yardstick: 13 in 36 in

pole: 18 ft x ft

[pic] [pic] [pic] x = 49.846

So the pole is about 50 feet tall.

Example 3: Suppose y varies directly with x and y = 12 when x = 9. Find an equation relating x and y, and sketch its graph.

Solution: Direct variation always produces a line that goes through the origin, so this line contains the points (0, 0) and (9, 12). Its slope is [pic], so the equation is [pic]. The graph is given on the right.

Problems

1. What is the constant of variation in the equation [pic]?

2. Which of the following equations represent direct variation? (More than one.)

a. [pic] b. [pic] c. [pic] d. [pic]

3. Which of the following graphs represent direct variation? (More than one.)

a. b

c. d.

4. Solve for x: [pic]

5. Suppose y varies directly with x, and y = 24 when x = 8. What is the value of y when x = 2?

6. Suppose y varies directly with x, and y = 4 when x = 9. What is the value of y when x = 12?

7. y varies directly as x. The following table shows values of x and y:

|x |3 |6 |P |

|y |7 |Q |35 |

What are the values of P and Q?

8. In the table on the right, x and y are linearly related.

a. What equation relates these variables?

b. Is this relation a direct variation? Why or why not?

9. A man stands near the Cape Hatteras Lighthouse in North Carolina. The shadow of the lighthouse is 83.2 ft long, and the man, who is 5.8 ft tall, casts a shadow that is 2.5 ft long. What is the height of the lighthouse?

10. If 3 cans of soup are on sale for $1.50, then how much should 2 cans of soup cost?

11. On a map, 30 miles are represented by 1.5 inches. How many miles are represented by 2 inches?

12. The cost of a certain metal varies directly as its weight. If 5 pounds cost $12, how much would 8 pounds cost?

13. A certain wheel travels a distance of 35 inches when it makes one revolution:

How far will a wheel that is twice as large travel when it makes one revolution?

14. When a 26 inch bicycle wheel makes one revolution, the bike travels a distance of 6.8 feet. How many revolutions will the wheel make when the bike travels a distance of 200 feet?

15. Marvin’s car takes 24 inch tires, but Marvin put 26 inch tires on it. When his speedometer reads 60 mph, how fast is Marvin really going?

16. Robert drove 336 miles on 12 gallons of gas. His tank holds 16.5 gallons. How much farther could he drive before running out of gas?

17. The number of calories a person burns performing an activity varies directly with the time the person spends performing the activity. A 140 pound person can burn 80 Calories by jogging for 20 minutes. How long should a 140 pound person jog to burn 100 Calories?

18. The graph on the right represents a direct variation. What is the constant of variation, and what equation relates x and y?

-----------------------

–2

–2

2

2

y

x

[pic]

Pole

Yard-stick

Shadow of yard-stick

Shadow of pole

Sun-ray

Tire size: 24 in 26 in

Speed: 75 mph x mph

–2

–2

y

x

x

y

x

y

x

10

0

y

–2

y

x

20

50

100

0

x y

1 1

2 3

4 7

7 13

35 in

?

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