Question 258806
A charter bus company has determined that the cost of providing x people a tour is
C(x) = 180.00 + 2.50x.

A full tour is 60 people. The cost of a ticket is $15.00 plus $0.25 for each unsold ticket.
 Determine 
(a) the revenue function,
r = x(.25(60-x)+15)
r = x(15 -.25x + 15)
r = x(30-.25x)
r = -.25x^2 + 30x
:
(b) the profit function,
p = r - c
p = -.25x^2 + 30x - (180+2.5x)
p = -.25x^2 + 30x - 180 - 2.5x
p = -.25x^2 + 27.5x - 180
:
(c) the company’s maximum profit,
Find the axis of symmetry on the above equation, x = -b/(2a); a=-.25, b=27.5
x = {{{(-27.5)/(2*-.25)}}}
x = 55 passengers for max profit
Find actual profit, replace x with 55
p = -.25(55^2) + 27.5(55) - 180
p = -.25(3025) + 1512.5 - 180
p = -756.25 + 1512.5 - 180
p = $576.25 profit at 55 passengers
:
(d) the number of ticket sales that yields the maximum profit.
We did that when we determined the profit; x = 55 passengers
:
It may be interesting to see this graphically
{{{ graph( 300, 200, -20, 80, -200, 1000, -.25x^2+30x, -.25x^2+27.5x-180) }}}
where 
x = no. of passengers
y = $$
Profit is green, revenue is purple. the difference between them represents the cost
:
Note that graph is not valid past x=60