Question

A community theater uses the function p(d)= -5d²+300d -200 to model the profit (in dollars) expected in a weekend when the tickets to a comedy show are priced at d dollars each: Part A: Write and solve an equation to find out the prices at which the theater would earn $2,300 in profit from the comedy show each weekend. Show your reasoning Part B: At what price would the theater make the maximum profit. and what is that maximum profit? Show your reasoning.​

Answer 1

• A. $66.33
• B. $4300

Explanation:

Given function:

• p(d)= -5d² + 300d - 200

Part A

p(d) = 2300, find d:

• 2300 = -5d² + 300d - 200
• 5d² - 300d + 200 - 2300 = 0
• 5d² - 300d - 2100 = 0
• d² - 60d - 420 = 0
• d²- 2*30d + 900 = 1320
• (d - 30)² = 1320
• d- 30 = √1320
• d = 30 + 36.33
• d = 66.33

Part B

Maximum point of the function is obtained at vertex, which is (x = -b/2a):

• d = - 300/-10 = 30

The maximum profit is:

• p(30) = -5(30)² + 300(30) - 200 = 4300