A rectangular garden will be built using a rock wall as one side and a fence for the other three...

1. TRANSLATE the problem information

• Given information:
  • \(\mathrm{A(x) = 80x - 2x^2}\) represents the area of the garden
  • \(\mathrm{x}\) represents the length of the two perpendicular sides (to the wall)
  • The graph intersects the x-axis at \(\mathrm{x = 0}\) and \(\mathrm{x = k}\) (positive)
  • We need to determine what k represents

2. INFER what an x-intercept means

• When a graph intersects the x-axis, the y-coordinate equals 0
• Since our graph shows \(\mathrm{y = A(x)}\), at \(\mathrm{x = k}\) we have \(\mathrm{A(k) = 0}\)
• This means k is the value of x where the area equals zero

3. SIMPLIFY to find the value of k

• Set the area function equal to zero: \(\mathrm{A(x) = 0}\)
• \(\mathrm{80x - 2x^2 = 0}\)
• Factor out 2x: \(\mathrm{2x(40 - x) = 0}\)
• This gives us \(\mathrm{x = 0}\) or \(\mathrm{x = 40}\)
• Since k is the positive x-intercept, \(\mathrm{k = 40}\)

4. INFER the physical meaning

• When \(\mathrm{x = 40}\) meters (perpendicular sides), the area is zero
• Check: If perpendicular sides are 40m each, that uses \(\mathrm{2(40) = 80m}\) of fence
• This leaves \(\mathrm{80 - 80 = 0}\) meters for the parallel side
• Result: A degenerate rectangle with zero area

Answer: D. The length, in meters, of the perpendicular sides that results in an area of 0 square meters.

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students may find \(\mathrm{k = 40}\) correctly but then confuse what this represents in context. They might think since 40 is a large number, it must represent the maximum area or the length that gives maximum area.

This confusion about interpreting the meaning of zero area may lead them to select Choice A (maximum area length) or Choice C (maximum area value).

Second Most Common Error:

Poor TRANSLATE reasoning: Students may misunderstand what the variable x represents in the function. If they think x is the parallel side length instead of the perpendicular sides, they'll misinterpret what k means entirely.

This misunderstanding of the variable definition may lead them to select Choice B (parallel side length) instead of recognizing it's about the perpendicular sides.

The Bottom Line:

This problem tests whether students can connect algebraic solutions (finding where a function equals zero) to real-world meaning (what it means for area to be zero in a geometry context). The key insight is that zero area occurs at a boundary condition where the rectangle becomes degenerate.