A farmer has 50 feet of fencing to enclose a rectangular plot adjacent to a river, using the river as...

1. TRANSLATE the problem setup

• Given information:
  • Total fencing available: 50 feet
  • Rectangular plot adjacent to river (river forms one side)
  • Need fencing for three sides only
  • x = length of sides perpendicular to river

• What this tells us: We need to express the constraint and find the area function

2. TRANSLATE the constraint equation

• Since fencing covers two widths (each length x) and one length parallel to river (call it y):
  \(2\mathrm{x} + \mathrm{y} = 50\)
• Solve for y: \(\mathrm{y} = 50 - 2\mathrm{x}\)

3. INFER the optimization strategy

• Area = length × width = \(\mathrm{x} \times \mathrm{y} = \mathrm{x}(50 - 2\mathrm{x})\)
• This gives us \(\mathrm{A(x)} = 50\mathrm{x} - 2\mathrm{x}^2 = -2\mathrm{x}^2 + 50\mathrm{x}\)
• Since this is a quadratic with negative leading coefficient, it has a maximum at its vertex

4. SIMPLIFY using the vertex formula

• For quadratic \(\mathrm{ax}^2 + \mathrm{bx} + \mathrm{c}\), vertex occurs at \(\mathrm{x} = -\mathrm{b}/(2\mathrm{a})\)
• Here: \(\mathrm{a} = -2, \mathrm{b} = 50, \mathrm{c} = 0\)
• \(\mathrm{x} = -50/(2(-2))\)
  \(\mathrm{x} = -50/(-4)\)
  \(\mathrm{x} = 50/4\)
  \(\mathrm{x} = 25/2\)

Answer: C) 25/2

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students often set up the constraint incorrectly as \(\mathrm{x} + \mathrm{y} = 50\) instead of \(2\mathrm{x} + \mathrm{y} = 50\), forgetting that there are TWO sides perpendicular to the river that need fencing.

Using \(\mathrm{x} + \mathrm{y} = 50\) leads to \(\mathrm{A(x)} = \mathrm{x}(50 - \mathrm{x}) = 50\mathrm{x} - \mathrm{x}^2\), giving vertex at \(\mathrm{x} = 25\).
This may lead them to select Choice E (25).

Second Most Common Error:

Inadequate INFER reasoning: Students may correctly set up the area function but fail to recognize they need to find the maximum. Instead, they might substitute answer choices or use trial-and-error rather than systematic optimization.

This leads to confusion and guessing among the answer choices.

The Bottom Line:

This problem tests whether students can properly translate a physical constraint (fencing three sides) into mathematics AND recognize the resulting optimization as a vertex-finding problem. The key insight is that "adjacent to river" means one side needs no fencing.