Chapter 11.3, Problem 67E

a.

To determine

To find: a quadratic model for the data using the regression feature of a graphing utility.

Answer to Problem 67E

  $y=0.12t^2-2.30t+\mathrm{25.09.}$

Explanation of Solution

Given information: The table shows the revenues y (in billions of dollars) for Goodyear

Tire & Rubber from 2011 through 2017.

Year t	Revenue , y (in billions of dollars)
2011	22.77
2012	20.99
2013	19.54
2014	18.14
2015	16.44
2016	15.16
2017	15.38

Calculation:

Let t represents the year, with t = 1 corresponding to 2011, so above table can be written as:

Year t	Revenue , y (in billions of dollars)
1	22.77
2	20.99
3	19.54
4	18.14
5	16.44
6	15.16
7	15.38

Using the regression feature of a graphing utility, the quadratic model for the above table data is:

  $y=0.12t^2-2.30t+\mathrm{25.09.}$

b.

To determine

To graph: the model found in part (a) using the graphing utility and estimate the slope of the graph when t = 4 and interpret the result.

Answer to Problem 67E

The slope at t =4 is about -1.34. In 2014, revenue for Goodyear Tire & Rubber was decreasing by about $1.34 billion per year.

Explanation of Solution

Given information:

Calculation:

The graph of the model $y=0.12t^2-2.30t+25.09$ , is given below.

  

The slope at t =4 is about -1.34. In 2014, revenue for Goodyear Tire & Rubber was decreasing by about $1.34 billion per year.

c.

To determine

To find: the derivative of the model in part (a) and then evaluate the derivative for t = 4.

Answer to Problem 67E

  $y\text{'}\text{at}t=4:m=-1.34$ .

Explanation of Solution

Given information:

Calculation:

  $\begin{array}{l}y\text{'}=\underset{h\to 0}{\lim }\frac{f(t+h)-f(t)}{h}\\ y\text{'}=\underset{h\to 0}{\lim }\frac{0.12{(t+h)}^2-2.30(t+h)+25.09-(0.12t^2-2.30t+25.09)}{h}\\ y\text{'}=\underset{h\to 0}{\lim }\frac{0.24th+0.12h^2-2.30h}{h}\\ y\text{'}=\underset{h\to 0}{\lim }0.24t+0.12h-2.30\\ y\text{'}=0.24t-2.30\\ y\text{'}\text{at}t=4:m=0.24(4)-2.30=-1.34\end{array}$