A small business models its weekly profit, P, in dollars, using the equation P = -0.3x^2 + 42x - 315,...
GMAT Advanced Math : (Adv_Math) Questions
A small business models its weekly profit, \(\mathrm{P}\), in dollars, using the equation \(\mathrm{P = -0.3x^2 + 42x - 315}\), where \(\mathrm{x}\) is the number of units it produces and sells. The fixed weekly costs represent the business expenses incurred even when zero units are produced. Based on this profit model, what are the fixed weekly costs for the business, in dollars?
0
42
315
357
1. TRANSLATE the problem information
- Given information:
- Profit equation: \(\mathrm{P = -0.3x^2 + 42x - 315}\)
- \(\mathrm{x}\) = number of units produced and sold
- Need to find: fixed weekly costs
- TRANSLATE what "fixed weekly costs" means: These are costs that occur even when zero units are produced, so we need P when \(\mathrm{x = 0}\).
2. INFER the solution approach
- To find costs when no units are produced, substitute \(\mathrm{x = 0}\) into the profit equation
- The result will show the profit (or loss) when production is zero
3. Substitute x = 0 into the profit equation
\(\mathrm{P = -0.3x^2 + 42x - 315}\)
\(\mathrm{P = -0.3(0)^2 + 42(0) - 315}\)
\(\mathrm{P = 0 + 0 - 315}\)
\(\mathrm{P = -315}\)
4. INFER the meaning of the result
- A profit of \(\mathrm{-315}\) dollars means a loss of \(\mathrm{\$315}\)
- This loss occurs when no units are produced, representing the fixed weekly costs the business must pay regardless of production level
Answer: C (315)
Why Students Usually Falter on This Problem
Most Common Error Path:
Weak TRANSLATE skill: Students misunderstand what "fixed weekly costs" means in the context of a profit function. They might think fixed costs are represented by one of the coefficients (like the constant term 315 or the linear coefficient 42) rather than understanding that fixed costs are revealed when production is zero.
This may lead them to select Choice B (42) or directly choose Choice C (315) without proper reasoning about what \(\mathrm{x = 0}\) represents.
Second Most Common Error:
Inadequate INFER reasoning: Students correctly substitute \(\mathrm{x = 0}\) and get \(\mathrm{P = -315}\), but fail to interpret what negative profit means. They might think the answer should be positive or get confused about the relationship between negative profit and fixed costs.
This leads to confusion and potentially selecting Choice A (0) or adding values to get Choice D (357).
The Bottom Line:
This problem tests whether students can connect business terminology ("fixed weekly costs") with mathematical function evaluation and properly interpret negative values in a real-world context. The key insight is recognizing that when a business produces nothing, any loss equals their fixed costs.
0
42
315
357