A hall charges $30 per person for a sports banquet when 120 people attend. For every 10 extra people that attend, the hall will decrease the price...

Let x be the number of people
attends.

Let the revenue be
y.

Let the initial revenue when 120 people attend is 30*120
= 3600

y= people attend *
price

y= (120+x)*(30-
1.5*(x/10)]

y=  (120+x) * ( 30 -
0.15x)

y= ( 3600 - 18x +30x -
0.15x^2)

==> y= -0.15x^2 +12x
+3600

Now we need to find the maximum
values.

First we will determine
y'.

==> y' = -0.3x
+12

Now we will find the critical
values.

==> -0.3x +12 =
0

==> -0.3x =
-12

==> x = -12/-0.3 =
40

The number of people that will maximize
the revenue is 40 plus the initial 120 people = 40+120 = 160
people