\(\mathrm{C(x) = 2x^2 - 40x}\)The function \(\mathrm{C(x)}\) represents a company's cost, in dollars, to produce x units of a product,...
GMAT Advanced Math : (Adv_Math) Questions
\(\mathrm{C(x) = 2x^2 - 40x}\)
The function \(\mathrm{C(x)}\) represents a company's cost, in dollars, to produce \(\mathrm{x}\) units of a product, where negative values indicate savings and positive values indicate costs. The graph of \(\mathrm{y = C(x)}\) in the xy-plane intersects the x-axis at 0 and \(\mathrm{b}\). What does \(\mathrm{b}\) represent?
The cost, in dollars, when 0 units are produced
The number of units that results in the minimum cost
The number of units that results in a cost of $0
The minimum cost, in dollars, that the company will have
1. TRANSLATE the key information
- Given: \(\mathrm{C(x) = 2x^2 - 40x}\) represents cost to produce x units
- The graph intersects the x-axis at 0 and b
- Need to find what b represents
Key insight: "intersects the x-axis" means the y-value (which is C(x)) equals 0.
2. TRANSLATE to set up the equation
Set \(\mathrm{C(x) = 0}\) to find x-intercepts:
\(\mathrm{2x^2 - 40x = 0}\)
3. SIMPLIFY by factoring
Factor out the common factor 2x:
\(\mathrm{2x(x - 20) = 0}\)
4. SIMPLIFY using zero product property
If \(\mathrm{2x(x - 20) = 0}\), then either:
- \(\mathrm{2x = 0}\), which gives \(\mathrm{x = 0}\)
- \(\mathrm{x - 20 = 0}\), which gives \(\mathrm{x = 20}\)
5. INFER the meaning in context
Since the x-intercepts are 0 and 20, we have \(\mathrm{b = 20}\).
Looking at what this means:
- x = number of units produced
- When \(\mathrm{x = 20}\), the cost \(\mathrm{C(20) = 0}\)
- So b represents the number of units that results in a cost of $0
Answer: C
Why Students Usually Falter on This Problem
Most Common Error Path:
Weak INFER skill: Students find \(\mathrm{b = 20}\) correctly but then misinterpret what this represents in the context. They might think that since we're looking at a cost function, b must represent some kind of cost value rather than recognizing it's the x-coordinate (number of units).
This confusion about what the variable represents may lead them to select Choice A or Choice D, both of which describe cost values rather than unit quantities.
Second Most Common Error:
Conceptual confusion about minimum vs. zero: Students might assume that where the cost equals zero must also be where the cost is minimized, since zero seems like it should be the "best" cost. They don't realize that the function continues to decrease beyond the x-intercept into negative values (savings), so the minimum occurs elsewhere.
This may lead them to select Choice B (The number of units that results in minimum cost).
The Bottom Line:
This problem tests whether students can distinguish between finding specific points on a graph (x-intercepts) and interpreting what those coordinate values mean in a real-world context. The key is recognizing that b is an x-coordinate representing units, not a y-coordinate representing cost.
The cost, in dollars, when 0 units are produced
The number of units that results in the minimum cost
The number of units that results in a cost of $0
The minimum cost, in dollars, that the company will have