SOLUTION: A new coffee shop sells various coffee related items. The profit of this new coffee shop can be described by the equation, P=-4n^2+64n-112, where P represents profit in thousand of

Question 1073754: A new coffee shop sells various coffee related items. The profit of this new coffee shop can be described by the equation, P=-4n^2+64n-112, where P represents profit in thousand of dollars and n represents the number of customers, in ten thousands.
a) What is the profit if no customers visit the store?
b) What are the break-even points of the store?
c)What is the maximum possible profit of the store?
d) If a company makes a $80 000 profit they receive an award. How many customers are needed in order for the company to receive the award?

Answer by htmentor(1343) (Show Source): You can put this solution on YOUR website!
a) If no customers visit, then n=0. P(0) = -112 (in thousands). The profit is -112,000, i.e. a loss of 112,000.
b) To break even, the company's net profit must be zero.
So we have -4n^2 + 64n - 112 = 0 -> n^2 - 16n + 28 = 0.
Factoring the LHS, we have (n-14)(n-2) = 0. This gives n = 2 and n = 14 (ten-thousands)
So the break-even points are n = 20,000 and n = 140,000 customers.
c) The maximum profit is obtained where dP/dn = 0 -> -8n + 64 = 0 -> n = 8 x 10,000 = 80,000 customers. So the profit is P(8) = $144,000.
d) 80 = -4n^2 + 64n - 112 -> -4n^2 + 64n - 192 = 0 -> n^2 - 16n + 48 = 0
Factoring the LHS, we get (n-12)(n-4) = 0.
There are two solutions: 40,000 and 120,000 customers.

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