An agriculturalist is managing an apple orchard that currently has 30 trees planted per acre, with each tree yielding an...
GMAT Advanced Math : (Adv_Math) Questions
An agriculturalist is managing an apple orchard that currently has 30 trees planted per acre, with each tree yielding an average of 500 apples. For each additional tree planted per acre, the average yield per tree is expected to decrease by 10 apples due to overcrowding. This relationship can be modeled by the equation \(\mathrm{Y = -10x^2 + 200x + 15,000}\), where \(\mathrm{x}\) is the number of additional trees planted per acre (\(\mathrm{x \geq 0}\)) and \(\mathrm{Y}\) is the total apple yield per acre. If this equation is graphed in the xy-plane, what value of \(\mathrm{x}\) corresponds to the maximum possible value of \(\mathrm{Y}\)?
1. TRANSLATE the problem information
- Given: \(\mathrm{Y = -10x^2 + 200x + 15,000}\) represents total apple yield per acre
- Find: The x-value that gives maximum Y
- Constraint: \(\mathrm{x \geq 0}\) (can't have negative additional trees)
2. INFER the mathematical approach
- This is a quadratic function in standard form \(\mathrm{y = ax^2 + bx + c}\)
- Since \(\mathrm{a = -10}\) is negative, the parabola opens downward
- A downward parabola has its maximum value at the vertex
- We need to find the x-coordinate of the vertex
3. SIMPLIFY using the vertex formula
- For \(\mathrm{y = ax^2 + bx + c}\), the vertex occurs at \(\mathrm{x = -b/(2a)}\)
- Here: \(\mathrm{a = -10}\), \(\mathrm{b = 200}\), \(\mathrm{c = 15,000}\)
- \(\mathrm{x = -b/(2a)}\)
\(\mathrm{= -200/(2(-10))}\)
\(\mathrm{= -200/(-20)}\)
\(\mathrm{= 10}\)
4. APPLY CONSTRAINTS to verify the solution
- \(\mathrm{x = 10}\) satisfies the constraint \(\mathrm{x \geq 0}\)
- This makes practical sense: 10 additional trees per acre
Answer: A (10)
Why Students Usually Falter on This Problem
Most Common Error Path:
Weak INFER reasoning: Students may not recognize this as a vertex-finding problem. They might try to substitute each answer choice into the equation to see which gives the highest Y value, turning it into a computational marathon instead of using the elegant vertex formula. This approach works but wastes valuable time and increases calculation error risk.
Second Most Common Error:
Poor SIMPLIFY execution: Students know to use \(\mathrm{x = -b/(2a)}\) but make sign errors. The most common mistake is \(\mathrm{x = -200/(2 \times 10)}\)
\(\mathrm{= -200/20}\)
\(\mathrm{= -10}\), forgetting that \(\mathrm{a = -10}\) (negative). This error may lead them to select Choice B (20) if they then take the absolute value, thinking "trees can't be negative."
The Bottom Line:
This problem tests whether students can quickly identify a quadratic optimization situation and efficiently apply the vertex formula. The agricultural context is just window dressing - the math is purely about finding where a parabola reaches its peak.