Chapter 1.4, Problem 64PS

Solution

To find: the price that the shop should charge to maximize monthly revenue and find the maximum monthly revenue.

Correct choice is A, that is, to maximize revenue, each Surfboard should be sold at $340.

Given information:

The shop sells 45 surfboards per month at price $500 each.

For each $20 decrease in price, about 5 more Surfboards per month are sold.

Property Used:

Factoring and Zeros:

To find the maximum or minimum value of a quadratic function, first use factoring to write the function in intercept form $y=a\left(x-p\right)\left(x-q\right)$ . Because the function’s vertex lies on the axis of symmetry $x=\frac{p+q}{2}$ , the maximum or minimum occurs at the average of the zeros p and q .

Calculation:

Let x represent the price decrease and $R\left(x\right)$ represents the monthly revenue.

Now, the required verbal model is:

Monthly revenue(dollars) = Number of Surfboards sold $\times$ price of each Surfboard

So, the quadratic equation is:

  $\begin{array}{l}R\left(x\right)=\left(45+5x\right)\times \left(500-20x\right)\\ R\left(x\right)=\left(5x+45\right)\left(-20x+500\right)\\ R\left(x\right)=5\left(x+9\right)\left[-20\left(x-25\right)\right]\\ R\left(x\right)=-100\left(x+9\right)\left(x-25\right)\end{array}$

Now, it is clear that here the zeros of the revenue function are $-9$ and 25.

The average of zeros is $\frac{25+\left(-9\right)}{2}=8$ .

So, to maximize revenue, each subscription should cost,

  $\begin{array}{c}500-20x=500-20\times 8\\ =500-160\\ =\$340\end{array}$

So, correct choice is A .