Chapter 1.2, Problem 84E

a.

To determine

To find the value of $\frac{f(2017)-f(2010)}{2017-2010}$ and interpret the result for the given problem.

Answer to Problem 84E

  $\frac{f(2017)-f(2010)}{2017-2010}$ =4.285

Explanation of Solution

Given information:

From the graph given ,

  $\begin{array}{l}f(2017)=93\\ f(2010)=63\end{array}$

Calculation:

  $\begin{array}{c}\frac{f(2017)-f(2010)}{2017-2010}=\frac{93-63}{2017-2010}\\ =\frac{30}{7}\\ =4.285\end{array}$

Since f(x) represents the sales of the company in the year x ,

  $\frac{f(2017)-f(2010)}{2017-2010}$ = 4.285 means that, average rate of change of sales between the years 2010 and 2017 is approximately $4.285 billion.

b.

To determine

To complete the given table

t	R(t)
0
1
2
3
4
5
6
7

Answer to Problem 84E

T	R(t)
0	62.3
1	72.3
2	80.472
3	86.822
4	91.356
5	94.06
6	94.94
7	93.996

Where , $R(t)=-0.912t^2+10.91t+62.3$

Explanation of Solution

Given information:

  $R(t)=-0.912t^2+10.91t+62.3$

  $R(t)=-0.912t^2+10.91t+62.3\text{ , 0}\le \text{t<<7}$ , where R is the revenue (in billions of dollars) and t=0 represents the year 2010.

Calculation :

  $\begin{array}{c}R(0)=-0.912(0)+10.91(0)+62.3\\ =62.3\end{array}$

  $\begin{array}{c}R(1)=-0.912(1)+10.91(1)+62.3\\ =72.3\end{array}$

  $\begin{array}{c}R(2)=-0.912(2^2)+10.91(2)+62.3\\ =80.472\end{array}$

  $\begin{array}{c}R(3)=-0.912(3^2)+10.91(3)+62.3\\ =86.822\end{array}$

  $\begin{array}{c}R(4)=-0.912(4^2)+10.91(4)+62.3\\ =91.356\end{array}$

  $\begin{array}{c}R(5)=-0.912(5^2)+10.91(5)+62.3\\ =94.06\end{array}$

  $\begin{array}{c}R(6)=-0.912(6^2)+10.91(6)+62.3\\ =94.94\end{array}$

  $\begin{array}{c}R(7)=-0.912(7^2)+10.91(7)+62.3\\ =93.996\end{array}$

Thus we can complete the table given in the question as :

t	R(t)
0	62.3
1	72.3
2	80.472
3	86.822
4	91.356
5	94.06
6	94.94
7	93.996

By comparing the results in the table and corresponding points in the given graph, we get t both are approximately same. That is, the mathematical model fits the data well.