Chapter 3.4, Problem 1MP

(a)

To determine

Write an inequality that models the boosters making a profit .

Answer to Problem 1MP

  $2.25x>325+0.15x$

Explanation of Solution

Given :

The athletic boosters for a local college raise money by selling popcorn at a concession stand. They charge customers $2.25 per box.

The popcorn machine and supplies (unpopped kernels, popping oil, and butter) are provided by a company that charges a fee of $250 per game plus $0.15 per box of popcorn sold. The boosters must supply their own empty boxes to fill. Empty boxes are sold in various quantities, as shown in the table below.

  

The athletic boosters buy four packages of 75 empty boxes to augment their supply of 40 that they already have on hand.

Calculation:

Profit is where net revenue exceeds total costs.

Let $x$ be the number of boxes of popcorn that the boosters sell at the next game.

Since, selling charges per box of popcorn = $2.25

So, the revenue earned by selling x boxes of popcorn = $2.25x$

Since , the athletic boosters buy four packages of 75 empty boxes to augment their supply of 40 that they already have on hand.

So, the total number of boxes they buy = $4\times 75=300$

The costs of 1 box of the package that has 75 empty boxes = $0.25

So, the total cost of 300 boxes = $300\times 0.25=\$75$

The popcorn machine and supplies (unpopped kernels, popping oil, and butter) are provided by a company that charges a fee of $250 per game plus $0.15 per box of popcorn sold.

So, the fees for x boxes sold = $250+0.15x$

Hence , total cost for the boxes = $250+0.15x+75=325+0.15x$

For the profit ,

Revenue > Total cost

So, the inequality that models the boosters making a profit , if the number of boxes of popcorn that the boosters sell at the next game :

  $2.25x>325+0.15x$

(b)

To determine

Solve your inequality from part (a). Interpret your solution in terms of the situation.

Answer to Problem 1MP

  $x\ge 155$

The number of boxes of popcorn that the boosters must sell at the next game to make a profit should be atleast 155.

Explanation of Solution

Calculation:

From part (a),the inequality that models the boosters making a profit , if the number of boxes of popcorn that the boosters sell at the next game :

  $\begin{array}{l}2.25x>325+0.15x\\ \Rightarrow 2.25x-0.15x>325+0.15x-0.15x\mathrm{..........}[\text{subtract }0.15x\text{ from each side]}\\ \Rightarrow 2.1x>325\\ \Rightarrow x>\frac{325}{2.1}\mathrm{......................................................}[\text{divide each side by 2}\text{.1]}\\ \Rightarrow x>154.76\\ \Rightarrow x\ge 155\end{array}$

So, the number of boxes of popcorn that the boosters must sell at the next game to make a profit should be atleast 155.