Chapter 4, Problem 15SGR

To determine

To find: the price which will maximize Sal’s T-shirt income.

Answer to Problem 15SGR

75 t-shirts at $\$15$ each.

Explanation of Solution

Given:

The price of each t-shirt is $\$10$ .

And the number of t-shirts sales in weekend is 100.

Sal estimate that 5 less shirt for each $1 increase.

Calculation:

According to the question,

Create an equation to represent the loss of 5 shirt sell for every $1 increase in price. Therefore,

  $\left(100-5x\right)\left(10+x\right)$ .

Now, using FOIL, to find the factor, thus

  $\begin{array}{l}\left(100-5x\right)\left(10+x\right)\\ =1000+100x-50x-5x^2\\ =-5x^2+50x+1000\end{array}$ .

Now, to find the maximum value, use the formula $x=-\frac{b}{2a}$ , which is used to find the axis of symmetry. Then,

  $\begin{array}{l}x=-\frac{b}{2a}\\ =-\frac{50}{2\left(-5\right)}\\ =\frac{50}{10}=5\end{array}$

Now, plug x back in to the original equation, then

  $\begin{array}{l}\left(100-5\left(5\right)\right)\left(10+5\right)\\ =\left(75\right)\left(15\right)\end{array}$

Here 75 represent the number of t-shirt sold, and 15 represent the cost.

Hence, Sal’s sales 75 t-shirt at $\$15$ each.