\(\mathrm{f(x) = -500x^2 + 25,000x}\) The revenue \(\mathrm{f(x)}\), in dollars, that a company receives from sales of a product is...
GMAT Advanced Math : (Adv_Math) Questions
\(\mathrm{f(x) = -500x^2 + 25,000x}\)
The revenue \(\mathrm{f(x)}\), in dollars, that a company receives from sales of a product is given by the function f above, where \(\mathrm{x}\) is the unit price, in dollars, of the product. The graph of \(\mathrm{y = f(x)}\) in the xy-plane intersects the x-axis at 0 and a. What does a represent?
1. TRANSLATE the problem information
- Given information:
- \(\mathrm{f(x) = -500x^2 + 25,000x}\) (revenue function)
- \(\mathrm{x}\) = unit price in dollars
- \(\mathrm{f(x)}\) = revenue in dollars
- Graph intersects x-axis at 0 and a
- What this tells us: When a graph intersects the x-axis, the function value equals 0
2. INFER the mathematical approach
- To find where the graph intersects the x-axis, we need to solve \(\mathrm{f(x) = 0}\)
- This will give us the x-values where revenue equals zero
3. SIMPLIFY the equation \(\mathrm{f(x) = 0}\)
- Set up: \(\mathrm{-500x^2 + 25,000x = 0}\)
- Factor out common terms: \(\mathrm{-500x(x - 50) = 0}\)
- This gives us: \(\mathrm{x = 0}\) or \(\mathrm{x = 50}\)
4. TRANSLATE back to the context
- The x-intercepts are at \(\mathrm{x = 0}\) and \(\mathrm{x = 50}\)
- Since we're told the intercepts are at 0 and a, then \(\mathrm{a = 50}\)
- This means when the unit price is $50, the revenue is $0
Answer: C. The unit price, in dollars, of the product that will result in a revenue of $0
Why Students Usually Falter on This Problem
Most Common Error Path:
Weak TRANSLATE skill: Students confuse what the variables represent or mix up x-intercepts with y-intercepts.
Some students think "a" represents the revenue value rather than the price value, or they think it represents the maximum revenue instead of a zero-revenue point. They might see the intersection point and think it's asking for the y-coordinate instead of what the x-coordinate means.
This may lead them to select Choice A (The revenue when unit price is $0) or Choice D (Maximum revenue).
Second Most Common Error:
Incomplete INFER reasoning: Students find that \(\mathrm{a = 50}\) but don't properly connect this back to the business context.
They might recognize that \(\mathrm{a = 50}\) is a unit price but incorrectly assume it must be the optimal price (maximum revenue point) rather than understanding it represents a break-even point where revenue is zero.
This may lead them to select Choice B (Unit price for maximum revenue).
The Bottom Line:
Success requires carefully translating between mathematical language (x-intercepts, function values) and business context (unit prices, revenue amounts), while understanding that x-intercepts represent input values that make the output zero.