Chapter 3.2, Problem 1MP

(a)

To determine

Write an inequality that relates the number of empty boxes the boosters need to buy for the next game, the number of empty boxes on hand, and the maximum number of boxes of popcorn the boosters expect to sell.

Answer to Problem 1MP

  $x+40\le 310$

Explanation of Solution

Given :

The boosters have 40 empty boxes on hand from the last game and will need to buy more for the next game. They know from previous experience that they can expect to sell no more than 310 boxes at each game.

Calculation:

The boosters have 40 empty boxes on hand from the last game.

Let

the number of empty boxes the boosters need to buy for the next game = x

So, the total number of empty boxes the boosters have in hand = x + 40

The maximum number of boxes the boosters can sell in each game is 310.

Hence , x + 40 should not exceed 310.

So, the inequality that relates the number of empty boxes the boosters need to buy for the next game, the number of empty boxes on hand, and the maximum number of boxes of popcorn the boosters expect to sell :

  $x+40\le 310$

(b)

To determine

Solve your inequality from part (a).

Answer to Problem 1MP

  $x\le 270$

Explanation of Solution

Given :

The boosters have 40 empty boxes on hand from the last game and will need to buy more for the next game. They know from previous experience that they can expect to sell no more than 310 boxes at each game.

Calculation:

From part (a) , the inequality that relates the number of empty boxes x the boosters need to buy for the next game, the number of empty boxes on hand, and the maximum number of boxes of popcorn the boosters expect to sell :

  $x+40\le 310$

Solve the inequality :

  $\begin{array}{l}x+40\le 310\\ \Rightarrow x+40-40\le 310-\mathrm{40..........}[\text{subtract 40 from each side]}\\ \Rightarrow x\le 270\end{array}$

So, the number of empty boxes the boosters need to buy for the next game should not exceed 270.

(c)

To determine

Find the packages of empty boxes that the athletic boosters can buy in order to have enough for the next game. (Consider that they might buy more than one package of empty boxes. Also note that once the boosters have enough empty boxes, they do not need to buy any additional packages.)

Answer to Problem 1MP

• 4 packages of 75 empty boxes per package.
• 1 package of 200 empty boxes per package and 1 package of 75 empty boxes per package.

Explanation of Solution

Given :

Empty boxes are sold in various quantities, as shown in the table below.

  

Calculation:

From part (b) , the number of empty boxes the boosters need to buy for the next game should not exceed 270.

So, they cannot buy the package that has 300 and 400 boxes, since it will exceed the required number of boxes and money spent will be excessive.

They can buy packages with 75 and 200 boxes per package in combination :

• 4 packages of 75 empty boxes per package.
• 1 package of 200 empty boxes per package and 1 package of 75 empty boxes per package.