A rectangular garden has a fixed perimeter of 66 feet. One side has length x feet. Let \(\mathrm{A(x)}\) denote the...

1. TRANSLATE the problem information

• Given information:
  • Rectangular garden with fixed perimeter of 66 feet
  • One side has length x feet
  • Need to find x that maximizes area \(\mathrm{A(x)}\)

2. INFER the constraint relationship

• Since it's a rectangle, if one side is x feet, the other side is some width w
• Using perimeter: \(2\mathrm{x} + 2\mathrm{w} = 66\)
• Strategic insight: Express everything in terms of x to create \(\mathrm{A(x)}\)

3. SIMPLIFY to find the width

• From \(2\mathrm{x} + 2\mathrm{w} = 66\):

\(2\mathrm{w} = 66 - 2\mathrm{x}\)

\(\mathrm{w} = 33 - \mathrm{x}\)

4. INFER the area function setup

• Area of rectangle: \(\mathrm{A} = \mathrm{length} \times \mathrm{width}\)
• Therefore: \(\mathrm{A(x)} = \mathrm{x}(33 - \mathrm{x}) = 33\mathrm{x} - \mathrm{x}^2 = -\mathrm{x}^2 + 33\mathrm{x}\)
• This is a quadratic function with negative leading coefficient (opens downward)

5. INFER the optimization strategy

• Since the parabola opens downward, the maximum occurs at the vertex
• For quadratic \(\mathrm{ax}^2 + \mathrm{bx} + \mathrm{c}\), vertex is at \(\mathrm{x} = -\mathrm{b}/(2\mathrm{a})\)

6. SIMPLIFY using vertex formula

• \(\mathrm{A(x)} = -\mathrm{x}^2 + 33\mathrm{x}\), so \(\mathrm{a} = -1, \mathrm{b} = 33\)

\(\mathrm{x} = -\mathrm{b}/(2\mathrm{a}) = -33/(2(-1)) = 33/2\)

Answer: B. \(33/2\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students struggle to recognize that they need to use the perimeter constraint to express the width in terms of x, creating a single-variable function.

Instead, they may try to work with two variables (x and w) without connecting them through the constraint equation. This leads to confusion about how to maximize area when both dimensions seem variable.

This causes them to get stuck and randomly select an answer.

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly set up \(\mathrm{A(x)} = -\mathrm{x}^2 + 33\mathrm{x}\) but make calculation errors when applying the vertex formula.

Common mistakes include: forgetting the negative sign in \(-\mathrm{b}/(2\mathrm{a})\), incorrectly identifying \(\mathrm{a} = -1\), or arithmetic errors like \(-33/(-2)\). These errors might lead them to select Choice A (11) or Choice D (33) instead.

The Bottom Line:

This problem tests whether students can connect multiple mathematical concepts: geometric constraints, function optimization, and quadratic vertex properties. The key insight is recognizing that optimization requires expressing area as a single-variable function using the given constraint.