Chapter 14, Problem 73RE

a.

To determine

Find the average rate of change of revenue from $x=25$ to $x=130$ wristwatches.

Answer to Problem 73RE

  $\boxed{\$61.29}$

Explanation of Solution

Given information:

The following data represent the revenue R (in dollars) received from selling $x$ wristwatches at Wilk’s Watch Shop.

  $\begin{array}{cc}Wristwatches,x& \mathrm{Re}venue,R\\ 0& 0\\ 25& 2340\\ 40& 3600\\ 50& 4372\\ 90& 6975\\ 130& 8775\\ 160& 9600\\ 200& 10,000\\ 220& 9900\\ 250& 9375\end{array}$

Find the average rate of change of revenue from $x=25$ to $x=130$ wristwatches.

Calculation:

The data in this table represents the total revenue R in dollars received from $X$ number of wristwatches at WW Shop.

  $\begin{array}{cc}Wristwatches,x& \mathrm{Re}venue,R\\ 0& 0\\ 25& 2340\\ 40& 3600\\ 50& 4372\\ 90& 6975\\ 130& 8775\\ 160& 9600\\ 200& 10,000\\ 220& 9900\\ 250& 9375\end{array}$

To find the average rate of change in revenue from $x=25$ to $x=130$ watches, consider the change in revenue over the change in watches, $\frac{\Delta R}{\Delta x}$ . We need to know the initial revenue and number of watches as well as the final revenue and number of watches. The initial number of watches is $x=25$ , the initial revenue is $R\left(25\right)=2,340$ , the final number of watches is $x=130$ , and the final revenue is $R\left(130\right)=8,775$ .

Plug these values into the average rate of change formula to find the average rate of change of revenue from $x=25$ to $x=130$ watches.

  $\begin{array}{l}\frac{\Delta R}{\Delta x}=\frac{R\left(130\right)-R\left(25\right)}{130-25}\\ =\frac{8,775-2,340}{105}\\ =\frac{6,435}{105}\end{array}$

Hence, the average rate of change of revenue from $x=25$ to $x=130$ watches is $\$61.29$ per watch.

b.

To determine

Find the average rate of change of revenue from $x=25$ to $x=90$ wristwatches.

Answer to Problem 73RE

  $\boxed{\$71.31}$

Explanation of Solution

Given information:

The following data represent the revenue R (in dollars) received from selling $x$ wristwatches at Wilk’s Watch Shop.

  $\begin{array}{cc}Wristwatches,x& \mathrm{Re}venue,R\\ 0& 0\\ 25& 2340\\ 40& 3600\\ 50& 4372\\ 90& 6975\\ 130& 8775\\ 160& 9600\\ 200& 10,000\\ 220& 9900\\ 250& 9375\end{array}$

Find the average rate of change of revenue from $x=25$ to $x=90$ wristwatches.

Calculation:

The data in this table represents the total revenue R in dollars received from $X$ number of wristwatches at WW Shop.

  $\begin{array}{cc}Wristwatches,x& \mathrm{Re}venue,R\\ 0& 0\\ 25& 2340\\ 40& 3600\\ 50& 4372\\ 90& 6975\\ 130& 8775\\ 160& 9600\\ 200& 10,000\\ 220& 9900\\ 250& 9375\end{array}$

To find the average rate of change in revenue from $x=25$ to $x=90$ watches, consider the change in revenue over the change in watches, $\frac{\Delta R}{\Delta x}$ . We need to know the initial revenue and number of watches as well as the final revenue and number of watches. The initial number of watches is $x=25$ , the initial revenue is $R\left(25\right)=2,340$ , the final number of watches is $x=90$ , and the final revenue is $R\left(90\right)=6,975$ .

Plug these values into the average rate of change formula to find the average rate of change of revenue from $x=25$ to $x=90$ watches.

  $\begin{array}{l}\frac{\Delta R}{\Delta x}=\frac{R\left(90\right)-R\left(25\right)}{90-25}\\ =\frac{6,975-2,340}{65}\\ =\frac{4,635}{65}\end{array}$

Hence, the average rate of change of revenue from $x=25$ to $x=90$ watches is $\$71.31$ per watch.

c.

To determine

Find the average rate of change of revenue from $x=25$ to $x=50$ wristwatches.

Answer to Problem 73RE

  $\boxed{\$81.40}$

Explanation of Solution

Given information:

The following data represent the revenue R (in dollars) received from selling $x$ wristwatches at Wilk’s Watch Shop.

  $\begin{array}{cc}Wristwatches,x& \mathrm{Re}venue,R\\ 0& 0\\ 25& 2340\\ 40& 3600\\ 50& 4372\\ 90& 6975\\ 130& 8775\\ 160& 9600\\ 200& 10,000\\ 220& 9900\\ 250& 9375\end{array}$

Find the average rate of change of revenue from $x=25$ to $x=50$ wristwatches.

Calculation:

The data in this table represents the total revenue R in dollars received from $X$ number of wristwatches at WW Shop.

  $\begin{array}{cc}Wristwatches,x& \mathrm{Re}venue,R\\ 0& 0\\ 25& 2340\\ 40& 3600\\ 50& 4372\\ 90& 6975\\ 130& 8775\\ 160& 9600\\ 200& 10,000\\ 220& 9900\\ 250& 9375\end{array}$

To find the average rate of change in revenue from $x=25$ to $x=50$ watches, consider the change in revenue over the change in watches, $\frac{\Delta R}{\Delta x}$ . We need to know the initial revenue and number of watches as well as the final revenue and number of watches. The initial number of watches is $x=25$ , the initial revenue is $R\left(25\right)=2,340$ , the final number of watches is $x=50$ , and the final revenue is $R\left(50\right)=4,375$ .

Plug these values into the average rate of change formula to find the average rate of change of revenue from $x=25$ to $x=50$ watches.

  $\begin{array}{l}\frac{\Delta R}{\Delta x}=\frac{R\left(50\right)-R\left(25\right)}{50-25}\\ =\frac{4,375-2,340}{25}\\ =\frac{2,035}{25}\end{array}$

Hence, the average rate of change of revenue from $x=25$ to $x=50$ watches is $\$81.40$ per watch.

d.

To determine

Find the quadratic function of best fit.

Answer to Problem 73RE

  $\boxed{R\left(x\right)=-0.25x^2+100.01x-1.24}$

Explanation of Solution

Given information:

The following data represent the revenue R (in dollars) received from selling $x$ wristwatches at Wilk’s Watch Shop.

  $\begin{array}{cc}Wristwatches,x& \mathrm{Re}venue,R\\ 0& 0\\ 25& 2340\\ 40& 3600\\ 50& 4372\\ 90& 6975\\ 130& 8775\\ 160& 9600\\ 200& 10,000\\ 220& 9900\\ 250& 9375\end{array}$

Using a graphing utility, find the quadratic function of best fit.

Calculation: Use a best fit function on a graphing utility to find the quadratic function the best fits the data in the table.

On a graphing utility, enter the data points found in the table by hitting STAT , 1: Edit, and then entering the $x$ values into L1 and the $R\left(x\right)$ values into L2. Then hit STAT, CLAC, 5: QuadReg, VARS, Y-VARS, 1:Function, 1: $Y_1$ and then hit enter. Then go back to the Y= screen, and the quadratic function of best fit should appear in the $Y_1$ spot.

Hence, the quadratic function that fits the data in the table the best is

  $R\left(x\right)=-0.25x^2+100.01x-1.24$

e.

To determine

Determine the instantaneous rate of change of revenue at $x=25$ wristwatches.

Answer to Problem 73RE

  $\boxed{\$87.51}$

Explanation of Solution

Given information:

The following data represent the revenue R (in dollars) received from selling $x$ wristwatches at Wilk’s Watch Shop.

  $\begin{array}{cc}Wristwatches,x& \mathrm{Re}venue,R\\ 0& 0\\ 25& 2340\\ 40& 3600\\ 50& 4372\\ 90& 6975\\ 130& 8775\\ 160& 9600\\ 200& 10,000\\ 220& 9900\\ 250& 9375\end{array}$

Using the function found in part (d), determine the instantaneous rate of change of revenue at $x=25$ wristwatches.

Calculation:

To find the instantaneous rate of change of revenue at $x=25$ watches, first find $R\left(25\right)$ by plugging in $x=25$ into the best fit quadratic function found in part (d)

  $R\left(x\right)=-0.25x^2+100.01x-1.24$

  $R\left(25\right)=-0.25{\left(25\right)}^2+100.01\left(25\right)-1.24$

So when $x=25$ , $R\left(25\right)=2,342.76$ . Plug these values into the derivative $R\text{'}\left(x\right)$ to find the instantaneous rate of change of the revenue at $x=25$

  ${\lim }_{x\to 25}\frac{R\left(x\right)-R\left(25\right)}{x-25}={\lim }_{x\to 25}\frac{-0.25x^2+100.01x-1.24-2,342.76}{x-25}$

= ${\lim }_{x\to 25}\frac{-0.25x^2+100.01x-1.24-2,344}{x-25}$

  $\begin{array}{l}={\lim }_{x\to 25}\frac{\left(x-25\right)\left(-0.25x+93.76\right)}{x-25}\\ ={\lim }_{x\to 25}(-0.2x+93.76)\end{array}$

  $\begin{array}{l}=-0.25(25)+93.76\\ =87.51\end{array}$

Hence, the instantaneous rate of change of revenue at $x=25$ watches is about $\$87.51$ per watch.