Film Night Ticket Target
A student council is running a film night in the school hall. The hall hire and equipment cost $650, and catering costs $3.20 per person. Adult tickets sell for $14 and student tickets for $9. The hall can hold 180 people. By Thursday, 64 adult tickets and 91 student tickets have been sold. The council wants a net profit of at least $1,200 to fund a shade area. The business manager suggests selling all remaining seats only as student tickets at the current price, because students are more likely to attend. The council must decide by Friday whether this plan can reach the target or whether it needs a different pricing or seating strategy. The hall cost cannot be reduced, no refunds are available, and no other costs apply. Ticket prices and costs include GST.
1. Write a net-profit expression in terms of adult tickets a and student tickets s, and calculate the profit from the tickets already sold. State whether the $1,200 target has been met.
Read the worked answer
Revenue is $14a + $9s$. Costs are $650 + $3.20(a+s)$. So net profit is
$P = 14a + 9s - 650 - 3.20(a+s) = 10.80a + 5.80s - 650$.
With $a=64$ and $s=91$:
$P = 10.80(64) + 5.80(91) - 650 = 691.20 + 527.80 - 650 = 569.00$.
The profit so far is $569. The target is $1,200, so the council is short by $1,200 - $569 = $631. The target has not been met.
2. The manager's plan is to sell all 25 remaining seats as student tickets. Using the profit expression, determine the maximum profit under this plan and whether it reaches the target. Show how many extra student tickets would be needed if capacity were not a limit.
Read the worked answer
The extra profit from one extra student ticket is the ticket price minus the catering cost: $9.00 - $3.20 = $5.80. Selling 25 extra student tickets adds $25 \times 5.80 = $145. Maximum profit under the plan is $569 + $145 = $714. This is still short of $1,200 by $486.
If $x$ extra student tickets could be sold, the condition is $5.80x \ge 631$. Thus $x \ge 631/5.80 = 108.79$, so at least 109 extra student tickets would be needed. Only 25 seats remain, so the manager's plan cannot reach the target.
3. Instead, assume all 180 seats are sold, with a adult tickets and 180 - a student tickets. Find the minimum number of adult tickets needed to reach $1,200 profit, and comment on whether the current sales pattern makes this realistic.
Read the worked answer
With all 180 seats sold,
$P = 10.80a + 5.80(180-a) - 650 = 10.80a + 1044 - 5.80a - 650 = 5.00a + 394$.
We need $5a + 394 \ge 1200$, so $5a \ge 806$ and $a \ge 161.2$. Since tickets are whole, at least 162 adult tickets are needed.
The council currently has 64 adult tickets sold and only 25 seats left. Even if all remaining seats were adult tickets, there would be only $64 + 25 = 89$ adult tickets. That is far below 162. The target is not realistic without changing prices, cutting costs, or increasing the audience, so the council should revise the plan rather than rely on selling the remaining seats as student tickets.