A landscaper is designing a rectangular garden. The length of the garden is to be 5 feet longer than the...

1. TRANSLATE the problem information

• Given information:
  • Length is 5 feet longer than width
  • Area is 104 square feet
  • Need to find the length

• What this tells us: We can use one variable (width) to express both dimensions, then use the area formula.

2. TRANSLATE the relationships into algebra

• Let \(\mathrm{w}\) = width of the garden (in feet)
• Then length = \(\mathrm{w + 5}\) (since it's 5 feet longer)
• Area = length × width, so: \(\mathrm{w(w + 5) = 104}\)

3. SIMPLIFY the equation

• Expand: \(\mathrm{w(w + 5) = w^2 + 5w = 104}\)
• Rearrange to standard form: \(\mathrm{w^2 + 5w - 104 = 0}\)
• Factor the quadratic: We need two numbers that multiply to -104 and add to 5
• Those numbers are 13 and -8: \(\mathrm{(w + 13)(w - 8) = 0}\)

4. Solve for the possible values

• From \(\mathrm{(w + 13)(w - 8) = 0}\):
• \(\mathrm{w = -13}\) or \(\mathrm{w = 8}\)

5. APPLY CONSTRAINTS to select the valid solution

• Since width must be positive in a real-world context: \(\mathrm{w = 8}\) feet
• Therefore, \(\mathrm{length = w + 5 = 8 + 5 = 13}\) feet

Answer: 13

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE reasoning: Students struggle to set up the relationship between length and width algebraically. They might write separate equations like "\(\mathrm{L = W + 5}\)" and "\(\mathrm{L \times W = 104}\)" but fail to substitute the first into the second to create a single-variable equation.

This leads to confusion about how to proceed and often results in guessing among answer choices.

Second Most Common Error:

Inadequate SIMPLIFY execution: Students may correctly set up \(\mathrm{w(w + 5) = 104}\) but struggle with factoring the resulting quadratic \(\mathrm{w^2 + 5w - 104 = 0}\). They might attempt to use the quadratic formula with calculation errors or give up on factoring.

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

The Bottom Line:

This problem requires students to bridge word relationships with algebraic manipulation. The key insight is recognizing that using one variable to express both dimensions creates a manageable single-variable quadratic equation.