Ticket Pricing with a Staffing Kink
The drama club currently charges $8 per ticket and averages 300 attendees per show. A student survey suggests that for each $1 increase in ticket price, attendance will drop by 25. The auditorium has 320 seats. Fixed costs are $1,200 per show, and variable costs are $2 per attendee. If attendance exceeds 250, the club must pay an extra $150 for additional staff. The club adviser wants to choose a ticket price that maximizes profit, but the principal has asked that the price not exceed $12 and that attendance remain at least 200. A parent volunteer proposes a $10 price, arguing that it is a round number and will look reasonable. The adviser asks for a mathematical recommendation with assumptions stated.
1. Build the profit function P(x) where x is the dollar increase over $8, including the piecewise staffing cost. Compute P(0), P(2), and P(3).
Read the worked answer
Attendance A=300−25x, price p=8+x, contribution per ticket p−2=6+x. Without staffing, profit is (6+x)(300−25x)−1200. Staffing adds $150 when A>250, i.e., 300−25x>250, so x<2. Thus P(x)=(6+x)(300−25x)−1350 for 0≤x<2, and P(x)=(6+x)(300−25x)−1200 for x≥2. P(0)=6·300−1350=1800−1350=450. P(2)=8·250−1200=2000−1200=800. P(3)=9·225−1200=2025−1200=825.
2. Find the price that maximizes profit subject to price≤$12 and attendance≥200. Show whether the unconstrained vertex is feasible and compare endpoints.
Read the worked answer
Constraints: 8+x≤12 gives x≤4; 300−25x≥200 gives x≤4. For x≥2, P=(6+x)(300−25x)−1200 = 1800+300x−150x−25x^2−1200 = 600+150x−25x^2. Vertex x=−150/(2·−25)=3, P=600+450−225=825. This is feasible: x=3≤4, and attendance 225≥200. Endpoint x=4: P=(10)(200)−1200=800. For 0≤x<2, P=(6+x)(300−25x)−1350=450+150x−25x^2, which increases on [0,2), so its best approach is at x approaching 2 giving 650, but at x=2 the staffing charge drops and P=800. Thus the maximum is at x=3: price $11, attendance 225, profit $825. The $10 proposal (x=2) yields $800, so it is $25 worse.
3. Critique the linear demand assumption near the $12 price cap and above it. What qualitative feature is missing, and how should that affect the recommendation?
Read the worked answer
The model assumes a constant 25-ticket drop per $1 at all prices. Real demand is likely nonlinear: near the maximum willingness to pay, a small increase may produce a larger proportional drop, while at low prices demand may be less sensitive. Above x=12 the linear model predicts zero attendance at $20, but actual demand may fall to zero sooner or later. The model also treats stated survey intent as actual purchase behavior. The recommendation at $11 is inside the observed range of price changes (up to $4) and is robust to modest curvature, but the adviser should present it as a model-based estimate, not a guarantee, and should test $10.50–$11.50 in a small presale before committing.