Question

A small cruising ship that can hold up to 66 people provides three-day excursions to groups of 42 or more. If the group contains 42 people, each person pays $58. The cost per person for all members of the party is reduced by $1 for each person in excess of 42. Find the size of the group that maximizes income for the owners of the ship.

Answer 1

50 people

Explanation:

Let x represent the number of people on the cruise. The amount they each must pay is ...

... ($58 -(x -42)) = $100 -x

The revenue from the group is the product of the number of people and the amount each pays:

... r(x) = x·(100 -x)

This describes a downward-opeing parabola with zeros at x=0 and x=100. The vertex (maximum) will be found halfway between those zeros, at x=50.

A group size of 50 maximizes revenue from the cruise.