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Name_______________________________________

Unit 2: Direct and Indirect Variation

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Review: Graphing Lines

Each equation below is in SLOPE-INTERCEPT form. Graph each equation using its slope and y-intercept.

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Each equation below is in STANDARD form. Graph each equation using its x- and y-intercepts.

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For each of the following sets of data, write the linear equation that best fits the data and interpret the meaning

of the slope and y-intercept of the line.

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Equation:

Meaning of slope:

Meaning of y-intercept:

2.

Equation:

Meaning of slope:

Meaning of y-intercept:

3.

Equation:

Meaning of slope:

Meaning of y-intercept:

4.

5.

Equation:

Meaning of slope:

Meaning of y-intercept:

5.

Equation:

Meaning of slope:

Meaning of y-intercept:

6.

Equation:

Meaning of slope:

Meaning of y-intercept:

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Linear, Quadratic, and Exponential Tables

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Linear, Quadratic, and Exponential Tables

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Categorizing Rates of Change GRAPH DESCRIPTION of RATE of CHANGE

Height Vs. Time

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Direct, Inverse, and Joint Variations Identify each of the following statements as a direct, inverse, or joint variation by filling in the blank with the words directly, inversely, or jointly.

1. Volume of a gas, V, at constant temperature varies ____________ with its pressure, P.

2. Intensity of sound varies ___________ with distance away from the object creating the sound.

3. The weight of a body varies ____________ with the square of the distance it is from the center of the earth.

4. The power of an electrical circuit varies ____________ as the resistance and current.

5. The heat loss through a glass window of a house on a cold day varies ____________ as the difference between the

inside and outside temperatures and the area of the window. The heat loss varies ____________ as the thickness

of the window glass.

6. The amount of sales tax paid varies ____________ as the total of the goods purchased.

7. The time to complete a job varies ____________ as the number of workers working.

8. To balance a seesaw, the distance a person is from the pivot is ____________ proportional to his/her weight.

9. The intensity of a light varies ____________as the square of the distance from the light source.

10. The time it takes to complete a specific trip varies ____________ as the speed of travel.

11. The cost of gas on a trip varies ____________ with the length of the trip.

12. The length of a spring varies ____________ with the force applied to it.

13. The number of congruent marbles that fits into a box is ____________ proportional to the cube of the radius of

each marble.

14. The number of people invited to dinner varies ___________ as the amount of space each guest has at the table.

15. The number of people invited to dinner varies ____________ as the number of pieces of silverware used.

16. The time it takes to harvest a crop varies ____________ with the number of people assisting in the harvest.

17. The time it takes a runner to complete a lap on the track varies ____________ as the speed of the runner.

18. The cost of a cake varies ____________as the cake’s thickness and the square of the radius.

19. The number of calories burned during exercise varies ____________ with the time spent performing the exercise.

20. The power generated by a windmill is ____________ proportional to the cube of the wind speed.

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Length, Height, and Time

The height of a platform and length of a ramp affect the time it takes to roll down the ramp.

a. For a fixed ramp length, how do you think the time it takes to ride down the ramp will change as

platform height increases?

b. Fr a fixed platform height. How do you think the time it takes to ride down the ramp will change as the

ramp length increases?

c. Suppose that one skateboard ramp is twice as long as another ramp. What relationship between

platform heights for those ramps do you think will allow skateboarders starting at the top of each ramp

at the same time to reach the bottom at the same time?

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Unit 1 Data Collection .5 Foot Height

Ramp Length

3 ft 4 ft 5 ft 6 ft 7 ft 8 ft

1st

Time (seconds)

2nd

Time (seconds)

3rd

Time (seconds)

4th

Time (seconds)

Time: (seconds)

.25 Foot Height

Ramp Length

3 ft 4 ft 5 ft 6 ft 7 ft 8 ft

1st

Time (seconds)

2nd

Time (seconds)

3rd

Time (seconds)

4th

Time (seconds)

Time: (seconds)

Examine the data from the two experiments. For each platform height, describe the relationship between roll time

and ramp length:

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8 Foot Length

Ramp Height

.25 ft

.5 ft

.75 ft

1.0 ft

1.25 ft

1.5 ft

Time: (seconds)

4 Foot Length

Ramp Height

.25 ft

.5 ft

.75 ft

1.0 ft

1.25 ft

1.5 ft

Time: (seconds)

Examine the data from the two experiments. For each a fixed ramp length, describe the relationship between roll

time and platform height:

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Direct Variation

EXAMPLE 1: The amount of money spent at the gas station varies directly with the number of gallons

purchased. When 11.5 gallons of gas was purchased the cost was $37.72. Determine k, the constant of variation

and the equation that represents this situation.

EXAMPLE 2: The number of bricks laid varies directly with the amount of time spent. If 45 bricks are laid in

65 minutes, determine the equation that represents this situation. Also determine the time it would take to lay

500 bricks.

EXAMPLE 3: The number of centimeters y in a linear measurement varies directly with the number of inches x

in the measurement. Pablo’s height is 152.4 centimeters or 60 inches. What is Maria’s height in centimeters if

she is 64 inches tall?

EXAMPLE 4: The number of gallons g of fuel used on a trip varies directly with the number of miles m

traveled. If a trip of 270 miles required 12 gallons of fuel, how many gallons are required for a trip of 400

miles?

EXAMPLE 5: Karen earns $28.50 for working six hours. If the amount m she earns varies directly with h the

number of hours she works, how much will she earn for working 10 hours?

EXAMPLE 6: A bottle of 150 vitamins costs $5.25. If the cost varies directly with the number of vitamins in

the bottle, what should a bottle with 250 vitamins cost?

EXAMPLE 7: Wei received $55.35 in interest on the $1230 in her credit union account. If the interest varies

directly with the amount deposited, how much would Wei receive for the same

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Indirect Variation

EXAMPLE 1: The time it takes to travel a fixed distance varies inversely with the speed traveled. If it takes

Pam 40 minutes to bike to the secret fishing spot at 9 miles per hour, what is the equation that represents this

situation? How long will it take if she rides 12 miles per hour?

EXAMPLE 2: The volume V of a gas kept at a constant temperature varies inversely as the pressure p.

If the pressure is 24 pounds per square inch, the volume is 15 cubic feet. What will be the volume when the

pressure is 30 pounds per square inch?

EXAMPLE 3: The time to complete a project varies inversely with the number of employees. If 3 people can

complete the project in 7 days, how long will it take 5 people?

EXAMPLE 4: The time needed to travel a certain distance varies inversely with the rate of speed. If it takes 8

hours to travel a certain distance at 36 miles per hour, how long will it take to travel the same distance at 60

miles per hour?

EXAMPLE 5: The number of revolutions made by a tire traveling over a fixed distance varies inversely to the

radius of the tire. A 12-inch radius tire makes 100 revolutions to travel a certain distance. How many

revolutions would a 16-inch radius tire require to travel the same distance?

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Joint Variation

EXAMPLE 1: Variable I varies jointly as the values of P and T. If I = 1200 when P = 5000 and T = 3 ,

find I when P = 7500 and T = 4.

EXAMPLE 2: The cost c of materials for a deck varies jointly with the width w and the length l. If c = $470.40

when w = 12 and l = 16 , find the cost when w = 10 and l = 25.

EXAMPLE 3: The value of real estate V varies jointly with the neighborhood index N and the square footage

of the house S. If V = $376, 320 when N = 96 and S = 1600, find the value of a property with N = 83 and S =

2150 .

EXAMPLE 4: The number of gallons g in a circular swimming pool varies jointly with the square of the radius

r2 and the depth d. If g = 754 when r = 4 and d = 2 , find the number of gallons in the pool when r = 3 and d =

1.5 .

EXAMPLE 5: The ideal gas law states that the volume V (in liters) varies directly with the number of

molecules n (in moles) and temperature T (in Kelvin) and varies inversely with the pressure P (in kilopascals).

The constant of variation is denoted by R and is called the universal gas constant. Estimate the universal gas

constant if V=256.1 liters; n=1 mole, T=288 K; P=9.5 kilopascals.

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Determine whether the following tables of data represent direct variation, indirect variation, or neither.

Determine the constant of variation (if applicable) and write the equation that generates the table.

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Find the Missing Variable:

1) y varies directly with x. If y = -4 when x = 2, find y when x = -6.

2) y varies inversely with x. If y = 40 when x = 16, find x when y = -5.

3) y varies inversely with x. If y = 7 when x = -4, find y when x = 5.

4) y varies directly with x. If y = 15 when x = -18, find y when x = 1.6.

5) y varies directly with x. If y = 75 when x =25, find x when y = 25.

Classify the following as direct, inverse, or neither.

6) m = -5p 9) c = 4

e 12) c = 3v

7) r = t

9 10) n = ½ f 13) u =

18

i

8) d = 4t 11) z = t

2.

What is the constant of variation for the following?

14) d = 4t 15) z = t

2. 16) n = ½ f 17) r =

t

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Answer the following questions.

18) If x and y vary directly, as x decreases, what happens to the value of y?

19) If x and y vary inversely, as y increases, what happens to the value of x?

20) If x and y vary directly, as y increases, what happens to the value of x?

21) If x and y vary inversely, as x decreases, what happens to the value of y?

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Classify the following graphs as direct, inverse, or neither.

22) 23)

24) 25)

Answer the following questions:

26) The electric current I, is amperes, in a circuit varies directly as the voltage V. When 12 volts are applied,

the current is 4 amperes. What is the current when 18 volts are applied?

27) The volume V of gas varies inversely to the pressure P. The volume of a gas is 200 cm3 under pressure of

32 kg/cm2. What will be its volume under pressure of 40 kg/cm

2?

28) The number of kilograms of water in a person’s body varies directly as the person’s mass. A person with a

mass of 90 kg contains 60 kg of water. How many kilograms of water are in a person with a mass of 50 kg?

29) On a map, distance in km and distance in cm varies directly, and 25 km are represented by 2cm. If two

cities are 7cm apart on the map, what is the actual distance between them?

30) The time it takes to fly from Los Angeles to New York varies inversely as the speed of the plane. If the trip

takes 6 hours at 900 km/h, how long would it take at 800 km/h?

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More Word Problems 1. For a fixed number of miles, the gas mileage of a car (miles/gallon) varies inversely with the number of

gallons used. One year an employee driving a truck averaged 24 miles per gallon and used 750 gallons of gas. If

the next year, to drive the same number of miles the employee drove a compact car averaging 39 miles per

gallon, how many gallons of gas would be used?

2. To build a sound wall along the highway, the amount of time t needed varies directly with the number of

cement blocks c needed and inversely with the number of workers w. A sound wall made of 2400 blocks, using

six workers takes 18 hours to complete. How long would it take to build a wall of 4500 blocks with 10 workers?

3. The time needed to paint a fence varies directly with the length of the fence and indirectly with the number of

painters. If it takes five hours to paint 200 feet of fence with three painters, how long will it take five painters to

paint 500 feet of fence?

4. The time to prepare a field for planting is inversely proportional to number of people who are working. A

large field can be prepared by five workers in 24 days. In order to finish the field sooner, the farmer plans to

hire additional workers. How many workers are needed to finish the field in 15 days?

5. An egg is dropped from the roof of a building. The distance it falls varies directly with the square of the time

it falls. If it takes 12 second for the egg to fall eight feet, how long will it take the egg to fall 200 feet?

6. The number of hours needed to assemble computers varies directly as the number of computers and

inversely as the number of workers. If 4 workers can assemble 12 computers in 9 hours, how many workers are

needed to assemble 48 computers in 8 hours?

7. The weight of a person varies inversely as the square of the distance from the center of the earth. If the radius

of the earth is 4000 miles, how much would a 180 pound person weigh, 2000 miles above the surface of the

earth?

8. The strength of a rectangular beam varies jointly as its width and the square of it depth. If the strength of a

beam three inches wide by 10 inches deep is 1200 pounds per square inch, what is the strength of a beam four

inches wide and six inches deep?

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Inverse, Direct, and Joint Variation

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Write the equation being described by each of the following statements. 21. The volume, v, of a balloon is directly proportional to the cube of the balloon’s radius, r._______________

22. The number, n, of grapefruit that can fit into a box is inversely proportional to the cube of the diameter, d,

of each grapefruit. __________________

23. The time, t, that a plane spends on the runway varies inversely as the take-off speed.__________________

24. The weight, w, that a column of a bridge can support varies directly as the fourth power of its diameter, d,

and inversely as the square of its length, l. __________________

25. The radiation, r, from the decay of plutonium is directly proportional to the mass, m, of the sample tested

and inversely proportional to the square of the distance, d, from the detector to the sample. ______________

Write an equation for and solve each of the following word problems. 30. The cost, c, in cents of lighting a 100-watt bulb varies directly as the time, t, in hours, that the light is on.

The cost of using the bulb for 1,000 hours is $0.15. Determine the cost of using the bulb for 2,400 hours.

31. The power, P, in watts of an electrical circuit varies jointly as the resistance, R, and the square of the

current, C. For a 240-watt refrigerator that draws a current of 2 amperes, the resistance is 60 ohms. What is

the resistance of a 600-watt microwave oven that draws a current of 5 amperes?

32. The force needed to keep a car from skidding on a curve varies directly as the weight of the car and the

square of the speed and inversely as the radius of the curve. Suppose a 3,960 lb. force is required to keep a

2,200 lb. car traveling at 30 mph from skidding on a curve of radius 500 ft. How much force is required to keep

a 3,000 lb. car traveling at 45 mph from skidding on a curve of radius 400 ft.?