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Question 1167856: 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 packages of various quantities, as shown in the table below. Remember that the boosters must purchase the empty boxes in these quantities. For example, if they only needed 20 boxes, they would still need to purchase 75 as that is the smallest amount they come in.
Number of boxes per package
75
200
300
400
Price per box
0.25
0.21
0.19
0.17
The boosters have 40 empty boxes on hand from the last game and will need to buy more for the next game against Oregon. They know from previous experience that they can expect to sell no more than 310 boxes at each game.
Part A: Determine how many new boxes the boosters need to purchase for the next game against Oregon.
Part B: Using the table above, determine how much money the boosters need to purchase the new boxes.
Part C: Set up and solve an inequality to determine how many boxes the boosters need to sell at the next game against Oregon to make a profit (break-even point).
I think I figured out how to calculate part A and B, but am having trouble formulating the inequality to solve.
Answer by KMST(5328)   (Show Source):
You can put this solution on YOUR website! They will need to buy at least 310-40=270 boxes. They should get a package of 300 boxes at 0.19 a box for a cost of 300x0.19=57, because ordering apackage of 200 and a package of 75 would cost more.
Their total costs depends of the number of boxes. Let's say they sell n boxes.
Their total cost would be 57+250+0.15n if tthey sell n boxes.
The proceds from the sale of those n boxes would be 2.25n.
If 2.25n=57+250+0.15n they are breaking even.
2.25n=57+250+0.15n -> 2.25n-0.15n=57+250 -> 2.10n=307 -> n=307/2.10=146.190467...
They cannot sell that number of boxes, because it is not a wnole number, but if they sell 147 they will turn a small (very small) profit.
As an inequality you would have that to turn a profit, n must be such that
2.25n>57+250+0.15n -> 2.25n-0.15n>57+250 -> 2.10n>307 -> n>307/2.10 n>146.190467... or n greater than or equal to 147.
with n=147 they collect 2.25x147=330.75 with total costs of
57+250+0.15x147=329.05 . That would be a profit of 330.75-329.05=1.70.
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