Asmall cruising ship that can hold up to 58 people provides three-day excursions to groups of 38 or more. if the group contains 38 people, each person pays $72. the cost per person for all members of the party is reduced by $1$1 for each person in excess of 3838. find the size of the group that maximizes income for the owners of the ship.

Let x be the number of persons

Let the price be 72 dollars person initially. 

Deduct 1 dollar for each person that is more than 38 persons.
72 - x, price for x added persons.

Revenue for the cruise ship company:

72 – x = 38+x

Let this be the revenue function: 

y = 72 – x * (38+x)

Find the maximum value. It will happen exactly in the middle between the two roots of (72 – x) * (38+x) = 0

Zeros are at 72 and -38. The independent value in the middle is:

72 + (-38) / 2 = 16

 

MAXIMUM REVENUE: y = (72 – 16) * (16 + 38) 3024 dollars.
NUMBER OF PEOPLE FOR THIS MAX IS x = 16

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