Chapter 4.1, Problem 55E

To determine

the sale price, if the owner wants to sell $4,000 worth of the large supreme pizzas next week.

Answer to Problem 55E

The sale price should be $10

Explanation of Solution

Given information: Marketing: Each week, Marino’s Pizzeria sells an average of 160 large supreme pizzas for $16 each. Next week, the pizzeria plants to run a sale on these large supreme pizzas. The owner estimates that for each 40^C decrease in price, the store will sell approximately 16 more large pizzas.

Calculation:

For each 40c decrease in the price, the store will sell approximately 16 more large pizzas

For each $1 decrease in the price, the store will sell 16/0.4 more large pizzas

For each $x decrease in the price, the store will sell $\frac{16x}{0.4}$ more large pizzas

The number of pizzas sold is expected to be $160+\frac{16x}{0.4}$

  $=160+40x$

The sale price of a pizza is 16 − x

The total sales target is the product of the sale price and the number sold

  $\begin{array}{l}\left(16-x\right)\left(160+40x\right)=4000\\ 2560+480x-40x^2=4000\end{array}$

Multiply both sides with − 1

  $40x^2-480x-2560=-4000$

Adding 4000 on both sides

  $40x^2-480x+1440=0$

Divide both sides by 40

  $x^2-12x+36=0$

Factorize

  ${\left(x-6\right)}^2=0$

Taking square root on both sides

  $\begin{array}{l}x-6=0\\ x=6\end{array}$

So the sale price should be

  $\$16-\$6=\$10$

The sale price should be $10