The function A defined by \(\mathrm{A(x) = 0.5x(80 - x)}\) gives the area, in square meters, of a rectangular region,...
GMAT Advanced Math : (Adv_Math) Questions
The function \(\mathrm{A}\) defined by \(\mathrm{A(x) = 0.5x(80 - x)}\) gives the area, in square meters, of a rectangular region, where \(\mathrm{0 \lt x \lt 80}\). In the function, \(\mathrm{x}\) represents the length of one of the sides in meters. For what value of \(\mathrm{x}\) is the area, \(\mathrm{A}\), a maximum?
1. TRANSLATE the problem information
- Given information:
- \(\mathrm{A(x) = 0.5x(80 - x)}\) represents area in square meters
- Domain: \(\mathrm{0 \lt x \lt 80}\)
- Need to find x that maximizes A(x)
2. INFER the mathematical approach
- This is a quadratic function optimization problem
- Since we need a maximum, we need to find the vertex of the parabola
- Two approaches available: vertex formula or symmetry method
3. INFER which approach to use
- I'll use the vertex formula approach, which requires standard form
- Expand \(\mathrm{A(x) = 0.5x(80 - x)}\) to get \(\mathrm{A(x) = -0.5x^2 + 40x}\)
4. SIMPLIFY to identify quadratic coefficients
- \(\mathrm{A(x) = 0.5x(80 - x)}\)
- \(\mathrm{A(x) = 40x - 0.5x^2}\)
- \(\mathrm{A(x) = -0.5x^2 + 40x}\)
- So \(\mathrm{a = -0.5, b = 40, c = 0}\)
5. INFER the parabola orientation
- Since \(\mathrm{a = -0.5 \lt 0}\), parabola opens downward
- Therefore it has a maximum (not minimum) at its vertex
6. SIMPLIFY using vertex formula
- For \(\mathrm{ax^2 + bx + c}\), vertex x-coordinate = \(\mathrm{-b/(2a)}\)
- \(\mathrm{x = -40/(2(-0.5))}\)
- \(\mathrm{x = -40/(-1)}\)
- \(\mathrm{x = 40}\)
Answer: B. 40
Why Students Usually Falter on This Problem
Most Common Error Path:
Weak INFER skill: Students don't recognize this as a quadratic optimization problem requiring the vertex. Instead, they might try to solve \(\mathrm{A(x) = 0}\) or substitute the answer choices to see which gives the largest area value. While substitution could work, it's inefficient and students often make calculation errors or don't test the right values systematically. This leads to confusion and guessing.
Second Most Common Error:
Poor SIMPLIFY execution: Students correctly identify the need to find the vertex but make arithmetic errors in the vertex formula calculation. For example, they might calculate \(\mathrm{x = -40/(2(-0.5)) = -40/(-1) = -40}\), forgetting that dividing by negative 1 gives positive 40. This may lead them to select Choice A (20) if they somehow get confused with the domain or think they need half the calculated value.
The Bottom Line:
This problem tests whether students can connect the real-world concept of "maximum area" to the mathematical concept of finding a quadratic function's vertex. The key insight is recognizing that optimization problems involving quadratic functions always involve finding the vertex.