A rectangular garden has a perimeter of 28 feet. If the length of the garden is x feet, where 0...

1. INFER the mathematical approach

• Given information:
  • Rectangular garden with perimeter 28 feet
  • Length = \(\mathrm{x}\) feet (where \(\mathrm{0 < x < 14}\))
  • Area function: \(\mathrm{A = x(14 - x)}\)
• Key insight: To maximize area, we need to find the maximum value of this quadratic function

2. SIMPLIFY to standard quadratic form

• Expand \(\mathrm{A = x(14 - x)}\):
  \(\mathrm{A = 14x - x^2 = -x^2 + 14x}\)
• This gives us standard form with:
  • \(\mathrm{a = -1, b = 14, c = 0}\)

3. INFER the optimization strategy

• Since \(\mathrm{a = -1 < 0}\), the parabola opens downward
• Maximum occurs at the vertex
• Use vertex formula: \(\mathrm{x = -b/(2a)}\)

4. SIMPLIFY using the vertex formula

• \(\mathrm{x = -b/(2a)}\)
  \(\mathrm{x = -14/(2(-1))}\)
  \(\mathrm{x = -14/(-2)}\)
  \(\mathrm{x = 7}\)

5. APPLY CONSTRAINTS to verify the solution

• Check domain: \(\mathrm{0 < 7 < 14}\) ✓
• The solution is valid

Answer: 7

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students make sign errors when applying the vertex formula, particularly with the negative coefficients.

They might calculate \(\mathrm{x = -14/(2(-1))}\) incorrectly as \(\mathrm{x = -14/2 = -7}\), forgetting to properly handle the double negative. Since \(\mathrm{-7}\) is outside the given domain (\(\mathrm{0 < x < 14}\)), this creates confusion about whether their approach is wrong or if they should use the absolute value, leading them to guess or select an incorrect answer.

Second Most Common Error:

Poor INFER reasoning: Students don't recognize this as a vertex-finding problem and instead try to solve by setting the derivative equal to zero or by trial-and-error testing values.

Without the systematic vertex approach, they may test a few values like \(\mathrm{x = 5}\), \(\mathrm{x = 10}\), etc., and pick whichever gives a larger area without finding the true maximum. This leads to selecting a reasonable but incorrect value.

The Bottom Line:

Success requires recognizing the optimization nature of the problem and systematically applying the vertex formula with careful attention to sign handling.