Inverse Variation

• Investigate inverse variation through real-world problems

• Learn the basic inverse variation equations• Graph inverse variation functions

In the past investigation you noticed that as one quantity increased the other quantity that was related to it also increased.

Name some of the relationships we have studied so far where this is true.

` Now think about this situation. Suppose you opened your classroom door by pushing on

it close to the hinges. Then suppose you opened the same door by pushing on

it farther from the hinges.

As the distance from the hinge increases, the force needed to open the door decreases. This is an example of an inverse relationship.

Speed versus TimePage 123Materials Needed

One CBRGraphing CalculatorConnecting Cables

Set up the course by marking a starting line and finish line 2.0 meters apart.

Be sure you have acquired the Inverse Program.Run the Inverse Program and follow the directions

on the calculator screen.To begin the walker stands at the starting line and

the CBR holder stands 1 meter behind the starting line, facing the walker.

The CBR holder presses the TRIGGER on the CBR and the walker starts walking about 1 second later.

The walker walks from the starting line to the finish line at a constant rate. The walker stops at the finish line and stands still until the 10 seconds is over.

When the walk is complete press ENTER.Isolate the part of the graph that shows the

walker moving. From the screen record the total time for

the walk and the average speed of the walker. Record this on the chart.

Switch jobs and collect additional information until you have 5 sets of data

Enter the total time in L1 and average speed in L2. Create a graph of the data.

Complete steps 7 and 8. Enter an equation in y1 that is of the form y=a/x that is a good model for the relationship between the total time and the average speed.

What does the value of a represent in this problem?

Return to the lists and enter L3=L1●L2. What do you notice about L3?

first walk total time•first walk speed second walk total time•second walk speed

Since

we can also write

first walk total time second walk speedsecond walk total time first walk speed

or

first walk total time second walk total timesecond walk speed first walk speed

Show that all three expressions are the same.

first walk total time•first walk speed second walk total time•second walk speed

first walk total time second walk speedsecond walk total time first walk speed

first walk total time second walk total timesecond walk speed first walk speed

How are the three equations different from the other proportions we were writing?

Tyline measured the force needed as she opened a door by pushing at various distances from the hinge. She collected the data in the table at the right. (N=Newtons, a metric measure of force.)

Enter the data in your calculator using L1 and L2.

Distance (cm)

Force (N)

40.0 20.9

45.0 18.0

50.0 16.1

55.0 14.8

60.0 13.3

65.0 12.3

70.5 11.6

75.0 10.7

Create a graph of the data.

Calculate the produce of the distance and force in L3. Find the average value of the products in L3.

Write the equation of the form y=a/x that fits this data.

Distance (cm)

Force (N)

40.0 20.9

45.0 18.0

50.0 16.1

55.0 14.8

60.0 13.3

65.0 12.3

70.5 11.6

75.0 10.7

After studying example B work with your group to complete problem 11 on page 129.

Be prepared to present your solution to the problem.

Elaine, Ellen, and Eleanor, who are identical triplets, were playing together on a seesaw. When two of them sat on one side and one on the other, some careful positioning was needed to make the seesaw balance. After playing, they came indoors and did an experiment to understand this relationship. They place a pencil under the center of a 12 inch ruler and gathered a pile of nickels. They placed 2 nickels 3 inches from the center on one side of the “seesaw.” Then they placed different numbers of nickels on the other side and moved them until the balance point was found. Here is the data.

Nickels 1 2 3 4 5 6

Distance to Center

6 3 2 1.5

1.2 1

Check the product of each data pair.

Nickels 1 2 3 4 5 6

Distance to Center

6 3 2 1.5

1.2 1

nickels•distance from center

If 5 nickels are placed 4 inches from the center of the balance, how far will 8 nickels need to be placed from the center of the balance to balance the ruler?

nickels•distance from center

If two of the triplets sit 7 feet from the center of the seesaw, how far from the center does the other triplet need to sit to balance the seesaw?

Evaluation of the dayPlease complete the short evaluation of

Day 2 of the workshop. Your input on what has taken place is

valued.