   Chapter 1.4, Problem 11E Calculus: Early Transcendental Fun...

6th Edition
Ron Larson + 1 other
ISBN: 9781285774770

Solutions

Chapter
Section Calculus: Early Transcendental Fun...

6th Edition
Ron Larson + 1 other
ISBN: 9781285774770
Textbook Problem

Automobile CostsThe data in the table show the variable costs of operating an automobile in the United States for 2000 through 2010, where t is the year, with t = 0 corresponding to 2000. The functions y1, y2 and y3 represent the costs in cents per mile for gas, maintenance, and tires, respectively. (Source: Bureau of Transportation Statistics) t y1 >2 y3 0 6.9 3.6 1.7 1 7.9 3.9 1.8 2 5.9 4.1 1.8 3 7.2 4.1 1.8 4 6.5 5.4 0.7 5 9.5 4.9 0.7 6 8.9 4.9 0.7 7 11.7 4.6 0.7 8 10.1 4.6 0.8 9 11.4 4.5 0.8 10 12.3 4.4 1.0 (a) Use the regression capabilities of a graphing utility to find cubic models for y1 and y3, and a quadratic model for y2.(b) Use a graphing utility to graph y1, y2, y3, and y1 + y2+ y3 in the same viewing window. Use the model to estimate the total variable cost per mile in 2014.

(a)

To determine

The cubic model for y1 and y3 and quadratic model for y2 by the use of regression capabilities of graphing utility of the table:

ty1y2y306.93.61.717.93.91.825.94.11.837.24.11.846.55.40.759.54.90.768.94.90.7711.74.60.7810.14.60.8911.44.50.81012.34.41.0

Explanation

For y1,

Use Ti83 to find the mathematical model as below:

Step1: Press STAT then 5 then ENTER to setup the calculator’s built-in list editor.

Step2: Press STAT and ENTER to select the list editor.

Step3: Enter the data. For L1 enter the X data, here it is the values of t and for L2 enter the Y data, here it is the value of y1 as shown below:

L1L206.917.925.937.246.559.568.9711.7810.1911.41012.3

Step4: Press STAT then CALC then 6 to select the “CubicReg”.

Step5: Press ENTER to confirm.

The calculator will display the equation as:

y=ax3+bx2+cx+da=0.0172b=0.305c=0.867

And,

d=7.31

Hence, the cubic model of the given data is y1=0.0172t3+0.305t20.87t+7.3.

For y3,

Use Ti83 to find the mathematical model as below:

Step1: Press STAT then 5 then ENTER to setup the calculator’s built-in list editor.

Step2: Press STAT and ENTER to select the list editor.

Step3: Enter the data. For L1 enter the X data, here it is the values of t and for L2 enter the Y data, here it is the value of y3 as shown below:

L1L201.711.821.831.840.750.760

(b)

To determine

To graph: The y1, y2 and y3 and y1+y2+y3 in the same window by the use of graphing utility, where, t is the year with t=0 corresponds to 2000 and the function y1, y2 and y3 represents the cost for mile for gas, maintenance and tires respectively. Also, find total variable cost per mile in 2014 by the use of model.

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