Polynomial word problem - maximizing profit

Question

A travel agency organizes adventure weekend for groups of 10 persons or more. The normal price per person for such a weekend is 500 euros. In order to attract larger groups, the agency adopts the following promotion strategy: if there are more than 20 participants in the group, then the price per person will be reduced for all participants wit 10 euros for every participant above the reference number 20. The travel agency has a fixed cost fo 3000 euros per weekend it organizes and a variable cost of 100 euros per participant.

a) Construct the function that models the agency's profit in terms of the group size, and sketch a graph of this function.

b) For which group size is the agency's profit at its largest? And, how large is the maximal profit then?

c) What is the upper limit on the number of participants the agency must set in order to avoid a deficit?

Answer 1

If I am understanding the idea:

P(n≤20) = 500n -3000 - 100n = 400n - 3000

P(n≥20) = (500 - 10(n-20))n -3000 - 100n = 600n -10n² -3000

This looks like a line connected to a concave-down parabola. The apex of the function would be the max.

You can complete the square, find where the derivative = 0, or graph and find the peak. (I got 30=n), then plug into the equation for the n that maximizes profit.

Good luck!