Question

A small cruising ship that can hold up to 70 people provides three-day excursions to groups of 44 or more. If the group contains 44 people, each person pays $72. The cost per person for all members of the party is reduced by $1 for each person in excess of 44. Find the size of the group that maximizes income for the owners of the ship.

Answer 1

Let, extra number of people are x.

it is given that cost per person for all members of the party is reduced by $1 for each person in excess of 44.

So, total cost C :

$C = ( 44+x)(72-x)$

Now, dC/dx = 0

$\dfrac{dC}{dx}=(72-x) + (44+x)(-1) =0\\\\72-x -44 -x =0\\\\2x =28 \\\\x = 14$

Total person on cruise 44 + 14 = 58 i.e less than the capacity.

Therefore, maximum income for the owner is for group of size 58.

Hence, this is the required solution.