A company's weekly profit, P(x), in dollars, is modeled by the function \(\mathrm{P(x) = -0.5x^2 + 200x - 5000}\), where...
GMAT Advanced Math : (Adv_Math) Questions
A company's weekly profit, P(x), in dollars, is modeled by the function \(\mathrm{P(x) = -0.5x^2 + 200x - 5000}\), where \(\mathrm{x}\) is the number of units the company produces and sells. What does the number 5000 represent in the function?
The maximum weekly profit, in dollars, for the company.
The weekly fixed costs, in dollars, for the company.
The number of units the company needs to sell to break even.
The profit earned per unit sold, in dollars.
1. TRANSLATE the problem information
- Given information:
- \(\mathrm{P(x) = -0.5x^2 + 200x - 5000}\) models weekly profit
- x = number of units produced and sold
- Need to determine what 5000 represents
2. INFER the approach
- To understand what a constant term means in a function, examine what happens when the variable equals zero
- When x = 0, we see what the company's situation is with no production/sales
3. SIMPLIFY by substituting x = 0
- \(\mathrm{P(0) = -0.5(0)^2 + 200(0) - 5000}\)
- \(\mathrm{P(0) = 0 + 0 - 5000}\)
- \(\mathrm{P(0) = -5000}\)
4. INFER the business meaning
- When no units are sold, profit = -$5000
- Negative profit means the company loses $5000
- Costs that occur even with zero production are called fixed costs
- Therefore, 5000 represents weekly fixed costs
Answer: B. The weekly fixed costs, in dollars, for the company.
Why Students Usually Falter on This Problem
Most Common Error Path:
Weak INFER skill: Students may not think to substitute x = 0 to understand the constant term's meaning. Instead, they might try to find the maximum of the parabola or look for break-even points, missing the straightforward interpretation of the constant term.
This leads to confusion about which mathematical feature corresponds to which business concept, causing them to guess between the remaining choices.
Second Most Common Error:
Conceptual confusion about profit vs. loss: Students might substitute x = 0 correctly and get -5000, but fail to recognize that negative profit represents costs or losses. They might think this negative value represents something unrelated to the business model.
This may lead them to select Choice A (maximum profit) by incorrectly assuming they need to find the vertex of the parabola instead.
The Bottom Line:
The key insight is that constant terms in real-world functions often represent baseline values - what happens when the variable is zero. In business contexts, this typically reveals fixed costs or initial conditions.
The maximum weekly profit, in dollars, for the company.
The weekly fixed costs, in dollars, for the company.
The number of units the company needs to sell to break even.
The profit earned per unit sold, in dollars.