Chapter 14, Problem 39RE

(a)

To determine

To find: The average rate of change of revenue from $x=25$ to $x=130$ wristwatches.

Answer to Problem 39RE

  $61.29$

Explanation of Solution

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

  $\begin{array}{cc}\text{Wristwatches, }x& \text{Revenue, }R\\ 0& 0\\ 25& 2340\\ 40& 3600\\ 50& 4375\\ 90& 6975\\ 130& 8445\\ 160& 9600\\ 200& 10000\\ 220& 9900\\ 250& 9675\end{array}$

The average rate of change is change in R over change in x.

Average rate of change $=\frac{8775-2340}{130-25}=\frac{429}{7}\approx 61.29$

(b)

To determine

To find: The average rate of change of revenue from $x=25$ to $x=90$ wristwatches.

Answer to Problem 39RE

  $72.32$

Explanation of Solution

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

  $\begin{array}{cc}\text{Wristwatches, }x& \text{Revenue, }R\\ 0& 0\\ 25& 2340\\ 40& 3600\\ 50& 4375\\ 90& 6975\\ 130& 8445\\ 160& 9600\\ 200& 10000\\ 220& 9900\\ 250& 9675\end{array}$

The average rate of change is change in R over change in x.

Average rate of change $=\frac{6975-2340}{90-25}=\frac{927}{13}\approx 72.32$

(c)

To determine

To find: The average rate of change of revenue from $x=25$ to $x=50$ wristwatches.

Answer to Problem 39RE

  $81.4$

Explanation of Solution

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

  $\begin{array}{cc}\text{Wristwatches, }x& \text{Revenue, }R\\ 0& 0\\ 25& 2340\\ 40& 3600\\ 50& 4375\\ 90& 6975\\ 130& 8445\\ 160& 9600\\ 200& 10000\\ 220& 9900\\ 250& 9675\end{array}$

The average rate of change is change in R over change in x.

Average rate of change $=\frac{4375-2340}{50-25}=\frac{407}{5}\approx 81.4$

(d)

To determine

To find: The quadratic best fit of the given data using graphing utility.

Answer to Problem 39RE

  $R=-0.2323x^2+96.129x+83.098$

Explanation of Solution

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

  $\begin{array}{cc}\text{Wristwatches, }x& \text{Revenue, }R\\ 0& 0\\ 25& 2340\\ 40& 3600\\ 50& 4375\\ 90& 6975\\ 130& 8445\\ 160& 9600\\ 200& 10000\\ 220& 9900\\ 250& 9675\end{array}$

Using graphing utility plot the points and draw graph.

  

The quadratic best fit of given data using graphing utility is $R=-0.2323x^2+96.129x+83.098$

Where, R is revenue when x number of wristwatches sold.

(e)

To determine

To find: The instantaneous rate of Revenue at $x=25$ .

Answer to Problem 39RE

  $84.51$

Explanation of Solution

Given: $R=-0.2323x^2+96.129x+83.098$

The Revenue function is $R=-0.2323x^2+96.129x+83.098$

First differentiate w.r.t x

  $R\text{'}=-0.4646x+96.129$

At $x=25$

  $R\text{'}=-0.4646\left(25\right)+96.129=84.51$

Hence, the instantaneous rate of Revenue is $84.51$ .