A small business manufactures custom phone cases. The daily profit P, in dollars, follows a quadratic function of the number...
GMAT Advanced Math : (Adv_Math) Questions
A small business manufactures custom phone cases. The daily profit \(\mathrm{P}\), in dollars, follows a quadratic function of the number of cases produced per day. The business achieves maximum daily profit of $288 when producing 18 cases per day and breaks even (zero profit) when producing 30 cases per day. Which equation represents the daily profit \(\mathrm{P}\), in dollars, when producing \(\mathrm{n}\) cases per day?
1. TRANSLATE the problem information
- Given information:
- Daily profit P follows a quadratic function of cases produced (n)
- Maximum profit: $288 when producing 18 cases per day
- Break-even point: $0 profit when producing 30 cases per day
- What this tells us: We have a downward-opening parabola with vertex at (18, 288) and a point at (30, 0)
2. INFER the appropriate form and strategy
- Since we know the maximum point (vertex), this calls for vertex form: \(\mathrm{P = a(n - h)^2 + k}\)
- The vertex \(\mathrm{(h,k) = (18, 288)}\), so we have \(\mathrm{P = a(n - 18)^2 + 288}\)
- We need to find parameter 'a' using the break-even condition
3. SIMPLIFY to find the unknown parameter
- Use the break-even point: when \(\mathrm{n = 30, P = 0}\)
- Substitute: \(\mathrm{0 = a(30 - 18)^2 + 288}\)
- Calculate: \(\mathrm{0 = a(12)^2 + 288}\)
- Simplify: \(\mathrm{0 = 144a + 288}\)
- Solve: \(\mathrm{144a = -288}\), so \(\mathrm{a = -2}\)
4. Write the final equation
- \(\mathrm{P = -2(n - 18)^2 + 288}\)
Answer: C
Why Students Usually Falter on This Problem
Most Common Error Path:
Weak INFER skill: Students don't recognize that the maximum point (18, 288) is the vertex, instead thinking they need to use standard form or trying to use (30, 0) as the vertex since it's mentioned last.
This confusion about which point represents the vertex may lead them to select Choice D (\(\mathrm{-2(n - 30)^2 + 288}\)) by incorrectly treating (30, 0) as the vertex.
Second Most Common Error:
Poor SIMPLIFY execution: Students set up the vertex form correctly but make arithmetic errors when solving \(\mathrm{144a = -288}\), perhaps getting \(\mathrm{a = 2}\) instead of \(\mathrm{a = -2}\).
This leads to an equation that doesn't match any answer choice, causing them to get stuck and guess.
The Bottom Line:
This problem tests whether students can distinguish between different types of key points on a parabola (vertex vs. other points) and apply the vertex form correctly. The key insight is recognizing that "maximum profit" signals the vertex location.