Precalc word problem

Question

A small cruising ship that can hold up to 70 people provides three-day excursions to groups of 48 or more. If the group contains 48 people, each person pays $64$. The cost per person for all members of the party is reduced by $1$ for each person in excess of 48. Find the size of the group that maximizes income for the owners of the ship.

Answer 1

The revenue the ship brings in would be the number of people multiplied by the price per person. For example, if the minimum 48 people went at the original $64 per person, the revenue would be 48 * 64 = $3072.

Let X = the number for additional people. Therefore the number of people going on the excursion would be represented by the expression 48 + X. Since for every additional person, the cost drops by one dollar, the cost per person would be represented by the expression 64 - X.

To calculate the revenue for X additional people, you'd multiply (48 + X)(64 - X) or 3072 + 16X - X².

This is a quadratic, so the maximum revenue occurs at the vertex of the parabola. Using the -b/2a formula, -16/(2*-1) = 8. 8 additional people will maximize the profit, therefore a group of 48 + 8 = 56 people should go on the excursion.