Chapter 1.1, Problem 92E

(a)

To determine

To graph: To sketch the graph of the given data of sales for Nike, Inc. from the year 2011 to 2017.

Explanation of Solution

Given information: The table below shows the sales y (in billions of dollars) for Nike, Inc. for each year x from 2011 through 2017, where $x=1$ represents 2011

Year x	Sales y
1	20.9
2	24.1
3	25.3
4	27.8
5	30.6
6	32.4
7	34.4

  

Interpretation: Graph of the sales for Nike, Inc. from 2011 to 2017 is linear and increasing.

That is the sales increases gradually from 2011 to 2017.

(b)

To determine

To determine the years in which the sales showed the greatest and least increase.

Answer to Problem 92E

The year in which the sales showed the greatest increase is 2011

The year in which the sales showed the least increase is 2012

Explanation of Solution

Given information: The table below shows the sales y (in billions of dollars) for Nike, Inc. for each year x from 2011 through 2017, where $x=1$ represents 2011

Year x	Sales y
1	20.9
2	24.1
3	25.3
4	27.8
5	30.6
6	32.4
7	34.4

Formula used: Slope between the points $\left(x_1,y_1\right)$ and $\left(x_2,y_2\right)$ is $\frac{y_2-y_1}{x_2-x_1}$

Calculation:

The slopes between the years can be calculated as:

  $s_{12}=\frac{y_2-y_1}{x_2-x_1}=\frac{24.1-20.9}{2-1}=3.2$

  $s_{23}=\frac{y_3-y_2}{x_3-x_2}=\frac{25.3-24.1}{3-2}=1.2$

  $s_{34}=\frac{y_4-y_3}{x_4-x_3}=\frac{27.8-25.3}{4-3}=2.5$

  $s_{45}=\frac{y_5-y_4}{x_5-x_4}=\frac{30.6-27.8}{5-4}=2.8$

  $s_{56}=\frac{y_6-y_5}{x_6-x_5}=\frac{32.4-30.6}{6-5}=1.8$

  $s_{67}=\frac{y_7-y_6}{x_7-x_6}=\frac{34.4-32.4}{7-6}=2$

The year in which the sales showed the greatest increase is between 2011 and 2012

(Since $s_{12}=3.2$ )

The year in which the sales showed the least increase is between 2012 and 2013

(Since $s_{23}=1.2$ )

(c)

To determine

Equation of the line between the years 2011 and 2017

Answer to Problem 92E

Equation of the line between the years 2011 and 2017 is $y=2.25x+18.65$

Explanation of Solution

Given information: Sales in the year 2011 is 20.9 and sales in the year 2017 is 34.4

Formula used: Equation of the line passing through the points $\left(x_1,y_1\right)$ and $\left(x_2,y_2\right)$ is

  $y-y_1=m\left(x-x_1\right)$ Where slope $m=\frac{y_2-y_1}{x_2-x_1}$

Calculation:

Slope between the years 2011 and 2017, $m=\frac{34.4-20.9}{7-1}=\frac{13.5}{6}=2.25$

∴Equation of the line between the years 2011 and 2017is

  $y-20.9=2.25\left(x-1\right)$

  $\Rightarrow y-20.9=2.25x-2.25$

  $\Rightarrow y=2.25x+18.65$

(d)

To determine

To interpret the slope of the line in part (c)

Answer to Problem 92E

Slope of the line is the average rate of change

Explanation of Solution

Given information: Slope between the years 2011 and 2017 is 2.25

Slope of the line is the average rate of change. Therefore, the rate of change of sales over the years 2011 and 2017 is 2.25

(e)

To determine

To estimate the sales in 2021

Answer to Problem 92E

Sales in 2021 is 43.4 billion dollars

Explanation of Solution

Given information: Equation of the line between the years 2011 and 2017 is $y=2.25x+18.65$ .......... (1)

Calculation:

When $x=11$ for the year 2021,

1. becomes

  $\begin{array}{c}y=2.25\left(11\right)+18.65\\ =24.75+18.65\\ =43.4\end{array}$

∴ Sales in 2021 is 43.4 billion dollars.

The estimate sales in the year 2021 using the formula in (1) may or may not be accurate.

Because the sales in the year 2021 depends on the sales in the previous years and other factors which cannot be determined using the equation.

That is, the actual answer can vary.