A small cruising ship that can hold up to 62 people provides three-day excursions to groups of 42 or more. If the group contains 42 people, each person pays $60. 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 shi

Question

Mathematics

Joshua

8 September, 08:54

A small cruising ship that can hold up to 62 people provides three-day excursions to groups of 42 or more. If the group contains 42 people, each person pays $60. 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 shi

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Theodore Mueller · Answer 1

51

Step-by-step explanation:

You can figure this out by making a simple function to represent the problem

This is the function you would get:

f (x) = (x + 42) (60 - x)

The x is the number of people above 42. so it the group contains just 42 persons, x is going to be 0 = each person is going to pay 60 so the total is going to be 2520

T he domain for x would be all integers from 0 to 18. Now all you have to do is determine which value of x will cause the value of f (x) to be the highest.

f (x) = (x + 42) (60 - x) = 60x-x^2+42*60-42x=-x^2+18x+2520

to find the highest value, you could find the maximum by derivating and equal to 0.

f' (x) = -2x+18=0--->x=9

So the size of the group that maximizes income for the owner is 51 persons