Regression Models: Activity B
Automobile Crumple Zones
(Source: Larson, Roland. Precalculus with Limits, 2d Ed.
Houghton-Mifflin, 1997. Pg. 237, pg. 281)
To protect the occupants of automobiles in accidents, the automobiles are designed with crumple zones. The crumple zones allow the occupants to move short distances when the automobiles come to abrupt stops. The force on the crash victim during impact is measured in g's. (One g is equal to the acceleration due to gravity.) In crash tests with vehicles moving at 90 km per hour, analysts measured the numbers y of g's that were undergone during deceleration by crash dummies that were permitted (by crumple zones and passive restraint systems) to move distances of x meters during impact.
The following is a sample of the data collected by the analysts. x (meters) 0.2
0.4
0.6
0.8
1 y (in g ’ s) 158 80 53 40 32
1. For this situation, what is a reasonable domain for x ?
2. Use a graphing utility to make a scatter plot of the data. Record your graph on the grid below. What does the plot tell you about this situation?
3. What mathematical models might be used to describe this data? Why?
What are some models that you would not choose? Why?
Charles A. Dana Center Mathematics TEKS Toolkit www.mathtekstoolkit.org

4. Use the regression capabilities of your graphing utility to fit each of the following models to the data. Determine the regression correlation coefficient, r , for each model. a. Natural Log Regression Model:
Y
1
= r = b. Natural Exponential Regression Model:
Y
2
= r = c. Power Regression Model:
Y
3
= r =
5. Another model for the data is y
4
= -3.00 + 11.88ln(x) + (36.94/x).
Graph each of the four models above in the same viewing window with the scatter plot.
Which model do you think is the best choice? Comment on accuracy and simplicity.
6. Use each of the four models to estimate the distance traveled during impact if the passenger deceleration must not exceed 30 g's. Describe how you do this with technology. Also do this for Y1 through Y3 mathematically .
Comment on the difference in the distances traveled given by the models.
What are the practical implications of choosing a model when designing the crumple zones?
Charles A. Dana Center Mathematics TEKS Toolkit www.mathtekstoolkit.org