Expert: Ahmed Salami - 5/31/2011

Question
Can you please help me with the following question? Thank you in advance! :) I greatly appreciate it.

A bus company will charter a bus that holds 50 people to groups of 35 or more. If a group contains exactly 35 people, each person pays $60. In larger groups, everybody's fare is reduced by $1 for each person in excess of 35. Determine the size of the group for which the bus company's revenue will be the greatest.

It's an optimization problem.

--Thank you again,
Ryner Lute

Answer
Hi Ryner,
Let x be the size in excess of 35 of the group. This means that each person pays $(60 - x) and there are (35 + x) of them. The company's revenue function is then;
R = (35 + x)(60 - x)
= 2100 + 25x - x²
The revenue is greatest when dR/dx = 0.
dR/dx = 25 - 2x
equating to zero,
25 - 2x = 0
x = 12.5
But of course there is no practical meaning of 12.5 people, for the purposes of payment. So we take either 12 or 13 and see which of them gives a greater revenue. Fortunately, in this situation we have symmetry and all points equidistant from 12.5 on either side give the same revenue. Its then up to the company to decide to take 12 or 13 people as they both generate the same revenue.

Regards