Question

A community theater sells about 150 tickets to a play each week when it charges $20 per ticket. For each $2 decrease on price about 25 more tickets per week are sold. Write a quadratic function to find how the theater can maximize weekly revenue. What should the theater charge to maximize revenue? What is the maximum weekly revenue? (6 pts) (Show all work.)

What is the equation:
What should the theater charge?
What is the maximum revenue?

Answer 1

Given two items with a price and quantity, this determines which is the better buy by comparing unit prices. Finds the better deal.

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1) Theoretical Method

2) Practical Method

3) Semi-Theoretical Method

Step-by-step explanation:

Answer 2

To maximize weekly revenue, the community theater should charge $11 per ticket, which would generate a maximum weekly revenue of approximately $2893. This is determined by creating a quadratic function based on the given ticket price and sales pattern and finding the vertex of the parabola.

Step-by-step explanation:

To determine the maximum weekly revenue and the ticket price that the community theater should charge, we can use a quadratic function. When the theater charges $20 per ticket, 150 tickets are sold, so the initial revenue (R) is $20 × 150 = $3000. If the price decreases by $2, 25 more tickets are sold, so if 'n' is the number of $2 decreases, the new price is $20 - 2n, and the number of tickets sold is 150 + 25n. Thus, the revenue function R in terms of 'n' is:

R(n) = (20 - 2n)(150 + 25n)

Expanding this equation we get:

R(n) = 3000 + 500n - 50n - 50n²

R(n) = -50n² + 450n + 3000

The graph of a quadratic function y = ax² + bx + c is a parabola, and the vertex of the parabola represents the maximum point if a < 0, which it is in this case (-50). We can find the vertex using the formula -b/(2a), which in this case would be:

n = -450/(2(-50)) = 4.5

Substituting n = 4.5 back into the price and tickets equations, we find the ticket price should be:

Price = $20 - 2(4.5) = $11

And the number of tickets sold would be:

Tickets = 150 + 25(4.5) = 262.5 (which we would likely round to 262 or 263)

The maximum revenue is then:

Maximum Revenue = $11 × 263 = $2893

Therefore, to maximize weekly revenue, the theater should charge $11 per ticket, which would lead to a maximum weekly revenue of approximately $2893.