Question

A community theater sells about 150 tickets to a play each week when it charges $20 per ticket.

Answer 1

Answer:

• r = 12.5p(32 -p)
• $16 per ticket
• $3200 maximum revenue

Step-by-step explanation:

The number of tickets sold (q) at some price p is apparently ...

  q = 150 + 25(20 -p)/2 = 150 +250 -12.5p

  q = 12.5(32 -p)

The revenue is the product of the price and the number of tickets sold:

  r = pq

  r = 12.5p(32 -p) . . . . revenue equation

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The maximum of revenue will be on the line of symmetry of this quadratic function, which is halfway between the zeros at p=0 and p=32. Revenue will be maximized when ...

  p = (0 +32)/2 = 16

The theater should charge $16 per ticket.

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Maximum revenue will be found by using the above revenue function with p=16.

  r = 12.5(16)(32 -16) = $3200 . . . . maximum revenue

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Additional comment

The number of tickets sold at $16 will be ...

  q = 12.5(32 -16) = 200

It might also be noted that if there are variable costs involved, maximum revenue may not correspond to maximum profit.