Chapter 3.4, Problem 28AYU

Which Model? An engineer collects the following data showing the speed $s$ of a Toyota Camry and its average miles per gallon, $M$.

(a) Using a graphing utility, draw a scatter diagram of the data, treating speed as the independent variable. What type of relation appears to exist between speed and miles per gallon?

(b) Based on your response to part (a), find either a linear model or a quadratic model that describes the relation between speed and miles per gallon.

(c) Use your model to predict the miles per gallon for a Camry that is traveling 63 miles per hour.

To determine

To calculate:

a. Graph a scattered diagram and determine the type of relation that exists between the 2 variables.

b. Find the equation of best fit.

c. Predict the miles per gallon for a Camry that is travelling 63 miles per hour.

Answer to Problem 28AYU

a. The graph is given below.

b. The equation of best fit is $y=1.229x+917.38$.

c. The Camry that travels 63 miles per hour will travel $27.08\text{ miles}$ per gallons.

Explanation of Solution

Given:

The given data represents the relation between the speed of a Toyota Camry and its average miles per gallon.

Formula used:

For a quadratic equation $y=a{x}^2+bx+c,$ if $a$ is positive, then the vertex $\left(\frac{-b}{2a},<mtext> </mtext>y\left(\frac{-b}{2a}\right)\right)$ is the minimum point and if $a$ is negative, the vertex $\left(\frac{-b}{2a},<mtext> </mtext>y\left(\frac{-b}{2a}\right)\right)$ is the maximum point.

Calculation:

We can draw the scatter diagram using Microsoft Excel.

Thus, on entering the $x$ and the $y$ values on excel, we have to choose the Scatter diagram form the insert option.

Then for getting the equation of best fit, we have to choose the Layout option and then Trendline and then more Trendline option. Then choose the option polynomial and then display the equation.

Thus, we get the scatter diagram with the equation of best fit as

a. The scatter diagram is drawn above and we can see that the given variables exhibit a linear relationship with positive slope.

b. The equation of best fit is $y=-0.0175x^2+1.9346x-25.341$.

c. When $x=63$ we get

$y=-0.0175{(63)}^2+1.9346(63)-25.341=27.0813$.

The Camry that travels 63 miles per hour will travel $27.08\text{ miles}$ per gallons.