EXERCISES 1.3

3. Yes, there is a linear relationship. There is a constant slope

of 0.5 between all data pairs.

c) The slope tells you that, on average, for every increase

of 100 pounds of car weight, the car will lose 0.73 miles

per gallon in city mileage. The y-intercept of 44.3 says

that if a car weighs zero pounds, it will average 44.3

miles per gallon of gas. In this context, the y-intercept

is meaningless.

d) The answers will vary depending on what a student

feels is the weight of a very small car and a very large

one. One reasonable answer might be between 1000

and 5000 pounds. The range would then vary from

f(10) to f(50) or between approximately 37 and 7 mpg.

e) Yes, the equation y = –0.74x + 44.3 appears to be a

good fit since the residuals are randomly scattered

with no apparent trend or pattern and are reasonably

small.

10. a) Using 1990 as year 0: (0, 442.2), (8, 158.5)

y = –35.46x + 442.2

b) More than likely what will happen in the future is that

sales of cassettes will decline and sales will level out.

It will probably never be zero sales. The linear model

predicts that in year 12.5 (2002), no cassettes will be

sold.

14. The correct function is (b) because it is the only one that

gives the correct values. For example, after 2 hours, the

vehicle has traveled 30 miles the first hour plus 65 miles

the second hour, for a total of 95 miles. Substituting 2 in

quation (a) gives 130 miles. Substituting 2 in equation (b)

gives 95 miles. Substituting 2 in equation (c) gives 160

miles.

8. All functions are decreasing, concave up, and pass

through the point (1, 1). None have intercepts.

15. a) Answers will vary depending on the value of A used.

Following are three sample answers, one using an

ordered pair associated with the group of smallest

diameters, a pair near the middle of the data, and one

near the upper end:

Using (19, 35), A =0.097

Using (28, 150), A =0.191

Using (39, 358), A =0.235

Another possibility is for students to use the average of

the ratios for A = 0.177.

The value of A makes some difference in the fit of the

data, but, each leaves a clearly increasing pattern, so

no matter what A is chosen, a second-degree power

function does not produce a good fit.

b) Since residuals are increasing, move up the Ladder of

Powers to find a curve that increases more rapidly.

c) The same ordered pairs as in Exercise 15(a) were used

to calculate three possible values for A. The following

graphs show the functions for