Venue Choice Under an Uncertain Headcount
A student organization has ₱18,000 for a two-hour event and must choose between two venues. Venue A charges a flat ₱5,000 plus ₱120 per attendee. Venue B charges ₱250 per attendee for the first 60 attendees, then ₱180 per attendee for each attendee beyond 60. The organizers expect 70 attendees but admit turnout could be as low as 30 if the final exam schedule changes. The budget must cover the venue only; any leftover goes to snacks. No deposit has been paid, and both venues allow cancellation without penalty until three days before the event.
1. Above what attendance does Venue A become the cheaper option? Show the comparison as a piecewise cost function.
Read the worked answer
Write each venue as a function of the number of attendees n.
- Venue A: A(n) = 5000 + 120n
- Venue B: B(n) = 250n when n ≤ 60; B(n) = 15,000 + 180(n − 60) = 4200 + 180n when n > 60
For n ≤ 60: set 5000 + 120n = 250n → 5000 = 130n → n ≈ 38.46. Since B has the steeper slope, B is cheaper below this point and A is cheaper above it. Because attendance must be a whole number, A is cheaper from n = 39 onward in this range.
For n > 60: set 5000 + 120n = 4200 + 180n → 800 = 60n → n ≈ 13.3, which is not in the interval, so there is no crossing point there. Since A's per-head rate (₱120) is lower than B's marginal rate (₱180), A stays cheaper throughout this range.
Conclusion: for any turnout of 39 or more, Venue A costs less; below 39, Venue B costs less.
2. At the expected 70 attendees, which venue should be booked, and how much is left for snacks?
Read the worked answer
Substitute n = 70 into both functions.
- A(70) = 5000 + 120(70) = 5000 + 8400 = ₱13,400
- B(70) = 15,000 + 180(10) = 15,000 + 1800 = ₱16,800
Venue A saves 16,800 − 13,400 = ₱3,400 at the expected turnout.
Budget left over:
- With Venue A: 18,000 − 13,400 = ₱4,600 for snacks
- With Venue B: 18,000 − 16,800 = ₱1,200 for snacks
Both fit inside the ₱18,000 budget, so the decision is not forced by the budget - it is forced by the cost gap. At the stated expectation of 70, Venue A is the correct booking.
3. The president argues, "Book Venue B - if only 30 people come, we lose less." Evaluate this reasoning and recommend a decision.
Read the worked answer
Check the low-turnout scenario the president is worried about.
- A(30) = 5000 + 120(30) = ₱8,600
- B(30) = 250(30) = ₱7,500
At 30 attendees Venue B is indeed ₱1,100 cheaper, so the president's arithmetic is correct. But the argument is incomplete in three ways:
- It optimises the unlikely case. The stated expectation is 70; the 30-attendee case is a contingency, not a forecast.
- The downside is bounded. Because both venues cost less than ₱18,000 across the whole plausible range (30 to 70), neither choice can break the budget. The worst-case penalty for choosing A is ₱1,100 - small next to the ₱3,400 saved at the expected turnout.
- The crossing point is known. Venue A only becomes cheaper at 39 attendees. The realistic risk is whether turnout falls below 39, not below 70.
Recommendation: book Venue A, since the expected turnout is well above the 39-attendee break-even and the penalty in the pessimistic case is modest. If the organizers judge that turnout below 39 is genuinely likely, book B instead - but they should state that probability before deciding, not after.