Concert ticket pricing with an attendance floor
The student council at a secondary school is setting the price for its annual concert. The hall holds 900 people. Fixed costs are $9,000, and each ticket costs the council an additional $6 to process and print. Last year the price was $20 and 660 tickets were sold. A student survey suggests that for every $2 increase in the ticket price, attendance will fall by 30, and for every $2 decrease, attendance will rise by 30. The council wants to cover costs but also wants to avoid pricing the concert so high that fewer than 300 students attend. They are considering prices between $22 and $38. The treasurer has proposed $25, while the events officer argues for $32 because 'a fuller hall creates better atmosphere.' The principal has asked for a written recommendation based on profit, not just revenue, and wants the attendance constraint respected.
1. Let p be the ticket price in dollars. Derive the attendance function and the profit function, and state the domain that respects both the council's price range and the minimum-attendance condition.
Read the worked answer
At p=20, n=660. The survey slope is -30 tickets per $2, or -15 per $1, so n(p)=660-15(p-20)=960-15p. The minimum attendance gives 960-15p ≥ 300, so p ≤ 44; the council's price range p ∈ [22,38] is therefore the binding domain.
Profit = revenue - variable cost - fixed cost = pn - 6n - 9000 = (p-6)(960-15p)-9000.
Expanding: P(p)= -15p^2+1050p-14760.
2. Find the price that maximises profit, compare the treasurer's $25 and the events officer's $32, and make a recommendation. Include the attendance and profit at the optimum.
Read the worked answer
Complete the square or use the vertex: P(p)=-15(p-35)^2+3615, so the maximum occurs at p=35. Attendance n=960-15(35)=435, which exceeds 300; profit = $3,615.
At $25: n=585, profit=(25-6)(585)-9000=11115-9000=$2,115. At $32: n=480, profit=(32-6)(480)-9000=12480-9000=$3,480.
Recommendation: charge $35 if the council can depart from the two named proposals; it is the feasible profit maximum. If forced to choose only between $25 and $32, choose $32 because it gives $1,365 more profit and still has 480 attendances. The $35 price is within the stated range and attendance floor, but the survey model should be treated as an estimate.