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Part 1: Creating Tables from Situations

One way to determine if a situation is an inverse variation is to create a table from the information given, and then multiply each pair to see if the products are equivalent.

Interactive exercise. Assistance may be required. Match the tables and the k values with the correct situations by dragging each one of them to the correct location.

How did you match each table with the situation?
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Check Your Answer
The values for x need to be reasonable for the situation. Transportation speeds, in miles per hour, could reasonably range from 10 to 40 miles per hour. The dimensions of a flower garden, in feet or yards, could reasonably include 3, 4, or 12 feet or yards. Savings accounts earn interest in percents.
How did you match each constant of variation, k, to the table and situation?
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Check Your Answer
Each pair of numbers, or numbers in the same row, of the table should have the same product. The product is equal to the value of k.

Part 2: Problem Solving Using Inverse Variation

The math problem of the day in Jeremy's math class is to determine whether or not the formula for the area and the formula for the perimeter of a rectangle with an area of 24 square centimeters are inverse variations.

Jeremy decided to create a plan, or algorithm, for answering the daily math question. Drag and drop the steps Jeremy decided to use in solving the question of the day in the correct order.

Interactive exercise. Assistance may be required. Drag each step to its proper position below.

Step 1: The formulas

Area of a rectangle: A = lw
Perimeter of Rectangle: P = 2l + 2w = 2(l + w)

Step 2: Create tables

Find the missing answer, and then click on the blank to check your answer.

Area

Length(cm)	Width (cm)	Area (sq cm)
1	24	Interactive button. Assistance may be required. _______ 24
2	12	Interactive button. Assistance may be required. _______ 24
3	8	Interactive button. Assistance may be required. _______ 24
4	6	Interactive button. Assistance may be required. _______ 24

Note: Since the table shows the possible length and width values for a rectangle with an area of 24 square centimeters, you have to use the same values in the table for the perimeter.

Perimeter

Length (cm)	Width (cm)	Length + Width (cm)	Perimeter P = 2(l + w) (cm)
1	24	Interactive button. Assistance may be required. _______ 25	Interactive button. Assistance may be required. _______ 50
2	12	Interactive button. Assistance may be required. _______ 14	Interactive button. Assistance may be required. _______ 28
3	8	Interactive button. Assistance may be required. _______ 11	Interactive button. Assistance may be required. _______ 22
4	6	Interactive button. Assistance may be required. _______ 12	Interactive button. Assistance may be required. _______ 24

Step 3: Are the formulas for the area and perimeter of the rectangle inverse variations?

The area of the rectangle is Interactive button. Assistance may be required. ______________, 24, the area of a rectangle Interactive button. Assistance may be required. _________ is an inverse variation.
The perimeter of the rectangle Interactive button. Assistance may be required. ___________, varies, the perimeter Interactive button. Assistance may be required. ____________ is not an inverse variation.

Conclusion: The formula for the area of a rectangle is an inverse variation, but the formula for the perimeter of a rectangle is not an inverse variation.

Pause and Reflect

Which of the relationships, area or perimeter of a rectangle, represents an inverse variation? How do you know?

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Check Your Answer

Area represents an inverse variation because the product of length and width, lw, is constant. Perimeter does not represent an inverse variation because the product of 2 and the sum of the length and width, 2(l + w), is not constant.

Practice:

The price per person to rent a limousine for a prom varies inversely as the number of passengers. The table below provides an example of this inverse variation. If it costs each couple $43.75, how many couples are renting the limousine together? How many people, not counting the driver, are in the limousine? Is this a reasonable number of people to have in a limousine?

Number of People	Cost Per Person
4	87.50
5	70.00
7	50.00

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Need a hint?

Determine the constant of variation by multiplying the number of people by the cost per person.

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Check Your Answer

First find the k value.

Find the product of one of the coordinates. Check your work below.
4 × 87.50 = 350.00
5 × 70.00 = 350.00
7 × 50.00 = 350.00
Since the k value is the product of two factors, divide $350.00 by $43.75 to find the number of couples.
350.00 ÷ 43.75 = 8
A couple is 2 people. To find the number of people, multiply the number of couples by 2.
8 × 2 = 16

Answers may vary but 16 people is a lot of people in a limousine; probably too many.

In the United States, electrical outlets have a voltage of 110 volts. According to Ohm’s Law, V = IR, where V represents the voltage, I represents the current in amperes, and R represents the resistance in ohms. Use this information to answer questions 2 and 3.

Complete the table below.

Current (I)
2.5

5

Resistance (R)

20

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Need a hint?
What is the constant of variation in this situation?
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Check Your Answer

Current (I)
2.5

5.5

5

Resistance (R)
44

20

22
Melanie is an engineer who designs electrical circuitry. She needs a circuit with 8 amperes of current. What resistance should she build into the circuit?
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Need a hint?
What is the constant of variation in this situation?
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Check Your Answer
13.75 ohms

Current (I)	2.5		5
Resistance (R)		20

Current (I)	2.5	5.5	5
Resistance (R)	44	20	22