Community Theater Ticket Pricing and Break-Even
The Riverside Community Theater has 320 seats: 120 floor seats and 200 balcony seats. For the spring production, fixed costs for rights, set construction, and rehearsal space are $4,000. Each ticket sold adds $7 in variable costs for processing, programs, and volunteer meals. The board plans to charge $28 for floor seats and $18 for balcony seats. Advance sales suggest they can sell 240 tickets, with the floor section selling out at 120 and the remaining 120 in the balcony. A donor has offered to underwrite a $4 student discount on balcony tickets, but only if the theater can still cover all costs at full capacity. The board wants to know the minimum number of tickets that must be sold at current prices, assuming the floor section sells out first, and whether the discount is financially safe.
1. Write the profit as a function of floor and balcony tickets sold, then evaluate the 240-ticket plan.
Read the worked answer
Let f be floor tickets and b balcony tickets. Revenue = 28f + 18b. Total cost = 4,000 + 7(f+b). Profit P = 28f + 18b - 4,000 - 7f - 7b = 21f + 11b - 4,000. For f=120 and b=120: P = 21(120) + 11(120) - 4,000 = 2,520 + 1,320 - 4,000 = -160. The plan loses $160, so it does not cover costs. The contribution per floor ticket is $21 and per balcony ticket is $11, which is why the ticket mix matters as much as the total count.
2. Find the minimum total number of tickets needed to break even at current prices, assuming the floor section sells out first.
Read the worked answer
With the floor sold out, f=120 and b = n - 120, where n is total tickets. Then P = 21(120) + 11(n - 120) - 4,000 = 2,520 + 11n - 1,320 - 4,000 = 11n - 2,800. Set P >= 0: 11n >= 2,800, so n >= 254.545. Since tickets are whole, n = 255. At 255 tickets, balcony = 135 and P = 2,520 + 1,485 - 4,000 = $5. So 255 is the minimum under this sales pattern, only $5 above break-even; any unsold floor seat or lower balcony sales would create a loss.
3. Evaluate the proposed $4 balcony discount at full capacity. Should the board accept it?
Read the worked answer
The discount makes the balcony price $14, so contribution per balcony ticket = 14 - 7 = $7. At full house f=120, b=200: P = 21(120) + 7(200) - 4,000 = 2,520 + 1,400 - 4,000 = -$80. So the discount is not safe at full capacity under current costs; it turns a $720 profit into an $80 loss. A $3 discount leaves balcony contribution at $8 and full-house profit = 2,520 + 1,600 - 4,000 = $120, still thin. The board should either cap the discount, raise the floor price, or secure added underwriting before promising the discount.