The perimeter P, in feet, of a rectangular garden is given by the expression 4s + 10, where s gt...

1. TRANSLATE the problem information

• Given information:
  • Perimeter: \(\mathrm{P = 4s + 10}\) feet
  • Width: \(\mathrm{w = s}\) feet
  • Need to find: expression for length

• What this tells us: We have a rectangle where we know the perimeter and width in terms of variable s, and we need the length in terms of s.

2. INFER the approach

• Since we know the perimeter and width, we can use the rectangle perimeter formula to find length
• Strategy: Use \(\mathrm{P = 2l + 2w}\), substitute known values, then solve for l

3. Set up the perimeter equation

Using the rectangle perimeter formula \(\mathrm{P = 2l + 2w}\):

\(\mathrm{4s + 10 = 2l + 2(s)}\)

4. SIMPLIFY to solve for length

First, distribute the 2:

\(\mathrm{4s + 10 = 2l + 2s}\)

Subtract 2s from both sides:

\(\mathrm{4s + 10 - 2s = 2l}\)

\(\mathrm{2s + 10 = 2l}\)

Divide both sides by 2:

\(\mathrm{l = (2s + 10) \div 2 = s + 5}\)

Answer: (A) s + 5

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students might misidentify what the problem is asking for or confuse which expression represents which dimension.

Some students might think the length is simply the other part of the perimeter expression, leading them to incorrectly reason that if width is s and perimeter is 4s + 10, then length must be related to the remaining terms. This may lead them to select Choice (B) (2s + 5) without properly using the perimeter formula.

Second Most Common Error:

Poor SIMPLIFY execution: Students might make algebraic errors when manipulating the perimeter equation.

For example, they might forget to divide the final step by 2, getting 2s + 10 as their final answer, or make distribution errors early on. This may lead them to select Choice (B) (2s + 5) or Choice (C) (s + 10).

The Bottom Line:

This problem requires students to systematically use the rectangle perimeter formula rather than trying to guess relationships between the given expressions. The key insight is recognizing that you must substitute into \(\mathrm{P = 2l + 2w}\) and solve algebraically.