Lesson 3.2

Revisiting Slope

Suppose you are taking a long trip in your car. At 5 P.M., you notice that the odometer reads 45,623 miles. At 9 P.M., you notice that it reads 45,831. You find your average speed during that time period by dividing the difference in distance by the difference in time.

45,831 45,6239 5

20852

4

miles milesaveragespeed

hours hours

milesmph

hours

Is 52 mi/h how fast you were driving most of the time between 5 P.M. and 9 P.M.?

The formula for the slope between two points, (x1, y1 ) and (x2, y2), is 2 1

2 1

y - yslope= ,where x1 x2

- xx

You can write the equation of a line as y=a + bx, where a is the y-intercept and b is the slope of the line.

Balloon BlastoffIn this investigation you will launch a rocket

and use your motion sensor’s data to estimate the rocket’s speed. Then you will write an equation for the rocket’s distance as a function of time. Choose one person to be the monitor and one person to be the launch controller.

Procedure Notes1. Make a rocket of paper and tape. Design

your rocket so that it can hold an inflated balloon and be taped to a drinking straw threaded on a string. Color or decorate your rocket if you like.

2. Tape your rocket to the straw on the string.

3. Inflate a balloon but do not tie off the end. The launch controller should insert it into your rocket and hold it closed.

4. Tie the string or hold it taut and horizontal.

Hold the sensor behind the rocket. At the same time the monitor starts the sensor, the launch controller should release the balloon. Be sure nobody’s hands are between the balloon and the sensor.

Retrieve the data from the sensor to each calculator in the group.

Graph the data with time as the independent variable, x. What are the domain and range of your data? Explain.

What’s a linear equation that describes the distance traveled by your rocket after it has been launched?

Sketch the graph of the data and select four representative points from the rocket data. Mark the points on your sketch and explain why you chose them.

Record the coordinates of the four points and use the points in pairs to calculate slopes. This should give six estimates of the slope.

Are all six slope estimates that you calculated in last step the same? Why or why not? Find the mean, median, and mode of your slope estimates. With your group, decide which value best represents the slope of your data. Explain why you chose this value.

What is the real-world meaning of the slope, and how is this related to the speed of your rocket?

1

0.941310 0.2768363.322

0.35 0.15m

2

1.744151 0.2768363.668

0.55 0.15m

3

2.394023 0.2768363.849

0.7 0.15m

4

1.744151 0.9413104.029

0.55 0.35m

5

2.394023 0.9413104.15

0.7 0.35m

6

2.394023 1.7441514.332

0.7 0.55m

Average slope = 3.89Median slope = 3.94

Slope means that the distance between the rocket and the sensor is increasing at 3.89 meters per second.

When one variable depends on the other variable, it is called the

dependent variable. The other variable is called the

independent variable. Time is usually considered an independent

variable.

ExampleDaron’s car gets 20 miles per gallon of gasoline. He

starts out with a full tank, 16.4 gallons. As Daron drives, he watches the gas gauge to see how much gas he has left.Identify the independent and dependent variables.Independent=miles Dependent = gallonsState a reasonable domain and range for this situation.Domain: 0≤x≤328 miles Range: 0≤y≤16 .4 gallonsWrite a linear equation in intercept form to model this

situation. Y=16.4 – 0.05xHow much gas will be left in Daron’s tank after he drives

175 miles?y=16.4-0.05(175) => y = 7.65 gallonsHow far can he travel before he has less than 2 gallons

remaining?2=16.4-0.05x => x=288 miles