Theatre Ticket Pricing with a Quadratic Demand Model
A small community theatre is planning ticket prices for its spring play. A survey suggests that if the ticket price is p dollars, the number of tickets sold will be approximately n(p)=300−12p, as long as 0≤p≤25. The theatre has 220 seats, so ticket sales cannot exceed capacity. The play costs $1,800 to produce, and the theatre keeps all ticket revenue. The board wants to choose a price that maximizes revenue, but it also must at least cover production costs. A board member proposes charging $16 per ticket, arguing that a higher price means more money per ticket. Another suggests $10, because 'more people will come.' The board asks for a mathematical analysis before voting. Tickets are sold only in whole numbers, but the board is treating the model as continuous for planning.
1. Write the revenue function R(p), state its domain taking the 220-seat capacity into account, and determine the price that maximizes revenue.
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Revenue is price times number sold: R(p)=p·n(p)=p(300−12p)=−12p²+300p. Capacity requires n(p)≤220: 300−12p≤220 ⇒ p≥80/12=20/3≈6.67. Also n(p)≥0 gives p≤25, and p≥0. So a realistic domain is 20/3≤p≤25. This is a quadratic with a=−12<0, so the vertex gives the maximum. p=−b/(2a)=−300/(2·−12)=300/24=12.50. R(12.50)=12.5(300−150)=12.5(150)=1875. At $12.50, predicted sales are 150 tickets, below the 220-seat capacity, so the maximum is feasible. Maximum revenue is $1,875.
2. Determine the range of ticket prices that cover the $1,800 production cost. Interpret the result in the board's context.
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Cover production cost: R(p)≥1800 ⇒ −12p²+300p≥1800. Divide by −12 and reverse the inequality: p²−25p+150≤0 ⇒ (p−10)(p−15)≤0 ⇒ 10≤p≤15. At p=10, R=10(180)=1800; at p=15, R=15(120)=1800; between these prices revenue is above 1800. So ticket prices from $10 to $15 inclusive cover the $1,800 cost. At $16, R=16(300−192)=16(108)=1728, so the $16 proposal loses $72 relative to production cost.
3. A board member says, 'If we charge $16, we only need to sell 113 tickets to cover costs, so we should.' Evaluate this reasoning using the demand model.
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The reasoning uses 1800/16=112.5, so at least 113 tickets would be needed at $16. But the demand model predicts only n(16)=300−192=108 tickets at that price, not 113. Revenue would be $1,728, below $1,800. The 'needed tickets' calculation ignores the effect of price on demand; the theatre cannot assume 113 sales at $16. A price near $12.50 maximizes revenue, and any price from $10 to $15 covers costs.