Chapter 12.3, Problem 73E

a.

To determine

To find: a quadratic model for the given data using the regression feature of a graphing utility. Let x represent the year, with x =6 corresponding to 2006.

Answer to Problem 73E

Quadratic model is $y=64.509x^2-724.08x+\mathrm{3045.8.}$

Explanation of Solution

Given information: The table shows the revenues y (in millions of dollars) for N from

2006 through 2015.

Year	Revenue y
2006	996.7
2007	1205.3
2008	1364.7
2009	1670.3
2010	2162.6
2011	3204.6
2012	3609.3
2013	4374.6
2014	5504.7
2015	6779.5

Calculation:

Let x represent the year, with x =6 corresponding to 2006.

Year x	Revenue y
6	996.7
7	1205.3
8	1364.7
9	1670.3
10	2162.6
11	3204.6
12	3609.3
13	4374.6
14	5504.7
15	6779.5

Using the graphing utility on above table data to find the model:

  $\begin{array}{c}\text{QuadReg}\\ y=a{x}^2+bx+c.\\ a=\mathrm{64.50871212.}\\ b=-\mathrm{724.0799242.}\\ c=\mathrm{3045.786818.}\end{array}$

  $y=64.509x^2-724.08x+\mathrm{3045.8.}$

b.

To determine

To graph : the model found in part (a) using graphing utility and estimate the slope of the graph when x =10 and give an interpretation of the result.

Answer to Problem 73E

  $m=\mathrm{566.1.}$

This means that the rate of change in the revenue in 2010 was $566.1 million.

Explanation of Solution

Given information: The table shows the revenues y (in millions of dollars) for N from

2006 through 2015.

Year	Revenue y
2006	996.7
2007	1205.3
2008	1364.7
2009	1670.3
2010	2162.6
2011	3204.6
2012	3609.3
2013	4374.6
2014	5504.7
2015	6779.5

Calculation:

Quadratic model is $y=64.509x^2-724.08x+\mathrm{3045.8.}$

The graph of this model using graphing utility is shown below

  

The slope at x =10 is:

  $\begin{array}{l}m=\underset{h\to 0}{\lim }\frac{f(10+h)-f(10)}{h}.\\ m=\underset{h\to 0}{\lim }\frac{64.509{(10+h)}^2-724.08(10+h)+3045.8-f(10)}{h}.\\ m=\underset{h\to 0}{\lim }\frac{64.509h^2+566.2h}{h}.\\ m=\underset{h\to 0}{\lim }(64.509h+566.1).\\ m=64.509(0)+\mathrm{566.1.}\\ m=\mathrm{566.1.}\end{array}$

This means that the rate of change in the revenue in 2010 was $566.1 million.

c.

To determine

To graph: the tangent line to the model when x =10 using graphing utility. Compare the slope given by the graphing utility with the estimate in part (b).

Answer to Problem 73E

The slop is the same as part (b).

Explanation of Solution

Given information: The table shows the revenues y (in millions of dollars) for N from

2006 through 2015.

Year	Revenue y
2006	996.7
2007	1205.3
2008	1364.7
2009	1670.3
2010	2162.6
2011	3204.6
2012	3609.3
2013	4374.6
2014	5504.7
2015	6779.5

Calculation:

The tangent line graph to the model when x =10 using graphing utility is shown below.

  

The slop is the same as part (b).