Unit 4 Lesson 1: INVERSE VARIATION Name ______

WARM-UP

Factor the following expressions completely.


THINK ABOUT THE SITUATION

Suppose you are going on a car trip with your family. You know you need to go 750 miles to get to your destination. What other piece of information would you need to know in order to figure out the average rate of the car during the entire trip?

What was your average rate if you travelled there in 25 hours? In 15 hours? In 10 hours?

What is happening to your average rate as your time decreases? Could you get there in no time at all?

What would your average rate be if you took 3 days to get there? What would it be if it took you a week? (Don’t forget to change your time into hours for your rate!)

INVERSE VARIATION NOTES

INVESTIGATION

We will use an experiment conducted by Galileo over 500 years ago to measure the effect of gravity on an object rolling downhill. It has been said that height and time vary inversely. Using the ramp materials, car and stopwatch, do the following experiment and fill in the chart that goes with it.

You will use a ramp of fixed length and compare the time T, that it takes the car to reach the end of the ramp as the height h, is changed.

EXPERIMENT DIRECTIONS

The car will be placedon the ramp and released when the timer gives the signal.The timer will time how long it takes for the car to get to the bottom of the ramp. This will be repeated 3 times.The recorder will write down the 3 times and calculate the average time for each height. We will assume that each book is one inch.

Number of Books / 2 / 3 / 4 / 5 / 6 / 7
1st time
2nd time
3rd time
Average Time it took to reach the bottom

Use the graph below to sketch your results from your table.

  1. What seems to be the pattern above between the change platform height and the roll time in each graph?
  1. What is happening to the time it takes to reach the bottom of the ramp as the height is increased?
  1. Do you think you can raise the ramp high enough to make the time go to zero? Why or why not?
  1. The highest the ramp could ever be is vertical. What would the time be if the ramp were vertical?
  1. It seems that as the ramp height increases, the time for the car to roll decreases. You hopefully realized in question 3 and 4 that the time could never be zero. However, is there a time the “roll time” is getting close to as you increase ramp height?
  1. How could you describe the time it would take for the car to get to the end of the ramp if its height were zero? Is this explainable? How could we word it in mathematics?
  1. According to our model is this an example of direct variation?
  1. It has been stated that height and speed vary inversely, so if your answer was no to the previous question, what went wrong? How could we fix this later?

Check to see if the tables below are Inverse Variation:

Extra Practice Word Problems

  1. y varies inversely as x. Given y = 4 when x = 2. Determine the inverse variation equation. Then determine y when x = 16.
  1. y varies inversely as x. y = 6 and x = 16. Determine the inverse variation equation. Then determine y when x = 4.
  1. The time, t, required to empty a tank varies inversely as the rate of r, of pumping. If a pump can empty a tank in 2.5 hours at a rate of 400 gallons per minute, how long will it take to empty a take at 500 gallons per minute?
  1. The force, F, needed to break a board varies inversely with the length, L, of the board. If it takes 24 pounds of pressure to break a board 2 feet long. How many pounds of pressure would it take to break a board that is 5 feet long?