Question 1003563
x, number of persons
Price is  76 dollars person initially.


Subtract 1 dollar for each person more than 40 persons.
{{{76-x}}}, price for x added persons.
-
-
Group cost, meaning revenue for the cruise ship company:
{{{highlight((76-x)(40+x))}}}
Treat this as a revenue function, y for convenience.


{{{highlight(y=(76-x)(x+40))}}}
This is factored.  Find the maximum value.  It will  happen exactly in the middle between the two roots of {{{(76-x)(x+40)=0}}}.



Zeros are at 76 and -40.  The independent value in the middle is  {{{(76+(-40))/2=18}}}.


MAXIMUM REVENUE:  {{{y=(76-18)(18+40)=3364}}} dollars.
NUMBER OF PEOPLE FOR THIS MAX IS  {{{highlight(x=18)}}}.