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

Algebra ->  Quadratic Equations and Parabolas  -> Quadratic Equations Lessons -> 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       Log On


   

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) About Me  (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.

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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
t+=+%28-b+%2B-+sqrt%28b%5E2+-+4ac%29%29%2F%282a%29

t+=+%28-%28-120%29+%2B-+sqrt%28%28-120%29%5E2+-+4%2810%29%2827%29%29%29%2F%282%2810%29%29

t+=+%28120+%2B-+sqrt%2813320%29%29%2F%2820%29

t+=+%28120+%2B+sqrt%2813320%29%29%2F%2820%29 or t+=+%28120+-+sqrt%2813320%29%29%2F%2820%29

t+=+11.77 or t+=+0.23
Each decimal value is approximate.


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

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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.