Question

A company charters a party boat that normally costs ​$56 per person. A group discount reduces the fare by ​$0.70 for each ticket sold. The maximum capacity of the boat is 80​ people, including the crew of 10 people. What size of group would maximize the boat​ company's revenue?

a) 30 people
b) 40 people
c) 50 people
d) 60 people

Answer 1

To determine the size of the group that would maximize the party boat company's revenue, we set up and derived a revenue function based on price and the discount for each additional ticket sold. The maximum revenue is achieved when 40 tickets are sold.

Step-by-step explanation:

The company charters a party boat with a maximum capacity of 80 people, including the crew of 10, so there are 70 tickets available to sell. As each additional ticket sold reduces the fare by $0.70, we can create a revenue function, R(x) = px - 0.70x^2, where x is the number of tickets sold and p is the starting price of $56. To maximize revenue, we take the derivative of the revenue function with respect to x and set it to zero to solve for x. We get R'(x) = 56 - 1.40x = 0, which simplifies to x = 40. Therefore, the group size that would maximize the boat company's revenue is 40 people (option b).