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.

Question

Mathematics

Cristopher Mccoy

23 March, 01:21

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.

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Rebecca Waller · Answer 1

50 people

Step-by-step 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.