SOLUTION: Dances at the community centre produce revenue R=-60t^2 +600t, where R is the revenue and t the ticket price in dollars. Francine, the manager, found that the expenses, C, for the

Question 1202278: Dances at the community centre produce revenue R=-60t^2 +600t, where R is the revenue and t the ticket price in dollars. Francine, the manager, found that the expenses, C, for the dances is modelled by C=162 - 120t. Note: Profit P = R- C.
a) Determine the equation to represent the profit.
b) Determine the break-even point (zero profit).
c) Find the maximum profit and the ticket price that yields this profit.
Answer by math_tutor2020(3816) (Show Source): You can put this solution on YOUR website!

Part (a)

Revenue = money coming in
Cost = money going out

Profit = Revenue - Cost
P = R - C
P = ( R ) - ( C )
P = ( -60t^2 +600t ) - ( 162 - 120t )
P = -60t^2 +600t - 162 + 120t
P = -60t^2 + 720t - 162 is the final answer.

===================================

Part (b)

The break-even point is when the company neither gains money nor loses money.

Set profit equal to zero to determine t.

P = 0
-60t^2 + 720t - 162 = 0
-6(10t^2 - 120t + 27) = 0
10t^2 - 120t + 27 = 0/(-6)
10t^2 - 120t + 27 = 0

Let's use the quadratic formula.
Plugging in a = 10, b = -120, c = 27

or

or
Each decimal value is approximate.

Answer:
Break-even point happens when t = 0.23 and when t = 11.77

===================================

Part (c)

The vertex in this case is the highest point. It corresponds to the max profit.

The x coordinate of the vertex is the midpoint of the roots.
Each root is a break-even point.

Average the two break-even points
(0.23+11.77)/2 = 6
The max profit happens when t = 6 is the ticket price.

Plug this into the profit function.
P = -60t^2 + 720t - 162
P = -60*6^2 + 720*6 - 162
P = 1998

Answer:
The max profit is $1,998. It occurs when the ticket price is $6.

RELATED QUESTIONS
A model for a company's revenue is {{{R = -15p^2 + 300p + 12000}}} where p is the price... (answered by nerdybill)

The weekly revenue R (in dollars) earned by a computer repair service is given by R=... (answered by ewatrrr)

The average nightly attendance at a certain movie theater is 100 when the ticket price is (answered by ewatrrr)

The revenue at the Assembly Center depends on the number of seats sold for the Willie... (answered by josmiceli)

A model for a company's revenue from selling a software package is r = -2.5p (squared)... (answered by stanbon)

hi, I was going through my math homework and i was wondering if you could help me with... (answered by josgarithmetic)

The cost, in dollars, to produce x widgets is C(x) = 3x + 32 , x ≥0. The... (answered by solver91311)

(Revenue) The John Deere company has found that the revenue from sales of heavy-duty... (answered by stanbon)

I need steps to solve this problem about maximum revenue.. Company's monthly revenue... (answered by ikleyn)