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Using Variables to Represent Quantities & How They Change

Together

Copyright © 2013 Carlson et al

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In the first two investigations we learned about variables and their usefulness for writing formulas to represent the relationship between two varying quantities. We also examined situations that led to our representing new quantities by using operations of addition, subtraction, multiplication, and division.

This investigation will provide you with additional practice on these skills, while reviewing the meaning of percent, and properties of a rectangle and circle.

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1. Write an expression to represent each of the following.

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a. 5 more than x

b. The sum of x and y

c. The amount remaining when 10 is reduced by x

d. How much greater x is than 10

e. A value that is 7 times as large as x

f. How many times as large 15 is compared to 4

g. 15 measured in units of 3.5

x + 5

x + y

10 – x

x – 10

7x

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2. Illustrate each quantity with a drawing and write an expression that describes the quantity. Your house and your car are 100 feet apart.

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a. The distance of your car from your house.

b. Your distance from your house as you walk x feet from your car toward your house. (Hint: Begin by making a drawing of your house, your car and yourself, and then label x on your drawing.)

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c. Your distance from your house as you walk y feet away from your car in a direction away from your house.

d. Your distance from your house as you walk z feet away from your house.

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e. The distance of your car from your house measured in units of 5 feet. (You need to know how many 5-feet long planks to purchase so that you can lay them between your car and house.)

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3. Recall that a percentage represents a part of 100. If the sales tax of an x dollar purchase is 8%, the sales tax is 0.08x or 8 cents for every 100 cents of merchandise purchased. Write an expression for each of the following.

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a. The total cost of an x dollar purchase if the sales tax is 7.5%.

b. The amount you pay for an item that regularly costs y dollars but is discounted by 15%

a. The price of a house that costs z dollars at the beginning of last year that has increased in price by 10% since the beginning of last year.

b. The total cost of a car that originally cost x dollars, and has been increased by 10% since it is in such high demand, plus the sales tax of 8%.

1.075x

0.85y

1.10z

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4. a. Construct and label the side lengths of 3 different rectangles that have a perimeter of 40 meters.

b. Describe the process you used to construct these 3 rectangles.

c. If the rectangle with a 40-meter perimeter has a width of 5 meters describe a process you could use to determine the rectangle’s length.

d. If the value of the rectangle’s width is 8 meters describe a process you could use to determine the rectangle’s length.

e. If the varying values of the rectangle’s width (in meters) are represented by w write a formula that defines the rectangle’s length l (in meters) in terms of the rectangle’s width w.

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5. The perimeter of a rectangle is 56 feet.

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a. Write a formula to represent the rectangle’s length l in feet in terms of its width w in feet.

b. The area of a rectangle A is determined by multiplying the length of the rectangle l by the width of the rectangle w (or A = (l)(w)). Write a formula that defines the area of a rectangle in terms of its width w (in feet) if the perimeter is 56 feet.

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5. The perimeter of a rectangle is 56 feet.

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c. What does the phrase, “express a rectangle’s length l in terms of its width w” mean to you?

d. What does the phrase, “express a rectangle’s Area A in terms of its width w” mean to you?

It means that you are to write l = <some expression with only a w>. The expression that l is equal to describes how to determine l when values of w are given.

It means that the goal is to define A = <some expression with a w>. The expression defines how to determine the area A when only the variable w is known.

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6. a. Label the following variables on the figure by identifying attributes of the below circle that represent the following quantities.

r = radius of the circle measured in feetd = diameter of the circle measured in feetC = circumference of the circle measured in feetA = area of the circle measured in square feet

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b. Write the formula to express a circle’s diameter in terms of its radius. Use the formula to determine the diameter of a circle that has a radius of 4.721 feet.

c. Write the formula to express a circle’s circumference in terms of its diameter. Use the formula to determine the circumference of a circle that has a diameter of 6.48 feet.

d. Use the formula to determine the circumference C of a circle in terms of its diameter d to explain how the length of a circle’s diameter and circumference are related.

Every circle’s circumference C is π (or approximately 3.14) times as long as the circle’s diameter d.

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e. Write the formula to express a circle’s circumference in terms of its radius. Use the formula to determine the circumference of a circle that has a radius of 3.5 feet.

f. Write the formula to express a circle’s area in terms of its diameter. Use the formula to determine the area of a circle that has a diameter of 3.5 feet.

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7. If you know the perimeter of a square is 20 feet, use this information to determine:

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a. the length of a side of the square.

b. the area of the square.

a. Construct a drawing of the square that has a perimeter of 20 feet. Label the sides and area on you drawing and represent the square’s area on your drawing.

The length of the side of the square is 5 feet since a square has 4 equal sides and ¼ of 20 is 5.

Since the area of a square A is given by the formula A = s2, and (5)2 = 25, so the area of the square with perimeter 20 feet is 25 square feet.

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