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

To find the size of the group that maximizes income for the owners of the ship, we can create a function for the revenue based on the number of people in excess of 48 and find the value that maximizes it.

Step-by-step explanation:

To find the size of the group that maximizes income for the owners of the ship, we need to determine the number of people that will result in the highest total revenue. Let's break down the problem step-by-step:

1. For a group of 48 people, each person pays $64.
2. For each person in excess of 48, the cost per person is reduced by $1.
3. We can represent the cost per person as $64 - ($48 - n) where n represents the number of people in excess of 48.
4. The total revenue can be calculated by multiplying the number of people by the cost per person.
5. To find the size of the group that maximizes income, we need to find the value of n that results in the highest revenue.
6. We can create a function, R(n), for the revenue based on the number of people in excess of 48.
7. To find the maximum revenue, we can take the derivative of R(n), set it equal to zero, and solve for n.
8. The value of n that maximizes income can then be added to 48 to get the size of the group.

Alternatively, we can create a table or graph to visualize the revenue based on the number of people in excess of 48 and find the maximum revenue that way.