Question

A small cruising ship that can hold up to 72 people provides three-day excursions to groups of 50 or more. If the group contains 50 people, each person pays $66. The cost per person for all members of the party is reduced by $1 for each person in excess of 50. Find the size of the group that maximizes income for the owners of the ship.

Answer 1

To maximize income for the ship owners, the optimal group size on the ship excursion is 83 people.

Step-by-step explanation:

Let's model the problem:

Let the number of additional people in the group be x.

Revenue = Number of people * Price per person

Price per person = Initial price - $1 for each additional person = $66 - $1x

Revenue = (50 + x)($66 - 1x)

Expand and simplify:

Revenue = $3300 + $65x - x^2

x^2 is a parabola that opens downwards

x is a positive number since we are adding people

Therefore, the revenue is maximized at the x -coordinate of the vertex.

Vertex = -(65/−2) = 32.5

Since we cannot have half a person, we round up to 33 people.

Final answer: 50 + 33 = 83

So, the number of the persons is 83.