A sail for a sailboat is in the shape of a right triangle. The height of the sail is 4...

1. TRANSLATE the problem information

• Given information:
  • Sail is a right triangle
  • Height is 4 feet greater than base length
  • Area is 126 square feet
  • Need to find the height

• What this tells us: We need to set up equations relating base, height, and area

2. INFER the approach

• Since we have one unknown relationship (height vs base) and one constraint (area), we can create a system
• Strategy: Define variables, write the relationship equation, then use the area formula

3. Set up the mathematical relationships

Let b = base length and h = height (both in feet)

From "height is 4 feet greater than base": \(\mathrm{h = b + 4}\)
From "area is 126 square feet": \(\mathrm{A = 126}\)

4. APPLY the area formula

For any triangle: \(\mathrm{Area = \frac{1}{2} \times base \times height}\)
So: \(\mathrm{126 = \frac{1}{2} \times b \times h}\)

5. SIMPLIFY by substitution

Substitute \(\mathrm{h = b + 4}\) into the area equation:
\(\mathrm{126 = \frac{1}{2} \times b \times (b + 4)}\)

Multiply both sides by 2 to eliminate the fraction:
\(\mathrm{252 = b(b + 4)}\)

Distribute on the right side:
\(\mathrm{252 = b^2 + 4b}\)

6. SIMPLIFY to standard quadratic form

Rearrange: \(\mathrm{b^2 + 4b - 252 = 0}\)

Factor this quadratic by finding two numbers that multiply to -252 and add to +4
Testing factors of 252: 18 and 14 work because \(\mathrm{18 \times 14 = 252}\)
For our equation: 18 and -14 work because \(\mathrm{(18)(-14) = -252}\) and \(\mathrm{18 + (-14) = 4}\)

So: \(\mathrm{(b + 18)(b - 14) = 0}\)

7. APPLY CONSTRAINTS to select the valid solution

From \(\mathrm{(b + 18)(b - 14) = 0}\), we get \(\mathrm{b = -18}\) or \(\mathrm{b = 14}\)

Since the base length cannot be negative: \(\mathrm{b = 14}\) feet

Therefore: \(\mathrm{h = b + 4 = 14 + 4 = 18}\) feet

Answer: C) 18

---

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skills: Students often struggle to correctly set up the relationship "height is 4 feet greater than base." They might write \(\mathrm{b = h + 4}\) instead of \(\mathrm{h = b + 4}\), essentially reversing which variable is larger.

With the incorrect relationship \(\mathrm{b = h + 4}\), when they substitute into \(\mathrm{126 = \frac{1}{2} \times b \times h}\), they get:
\(\mathrm{126 = \frac{1}{2} \times (h + 4) \times h}\), leading to \(\mathrm{252 = h^2 + 4h}\), or \(\mathrm{h^2 + 4h - 252 = 0}\)

This gives the same quadratic equation but in terms of h instead of b. Factoring gives \(\mathrm{h = 14}\) or \(\mathrm{h = -18}\). Since height must be positive, they get \(\mathrm{h = 14}\), leading them to select Choice A (14).

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly set up the equations but make algebraic errors when expanding \(\mathrm{b(b + 4)}\) or when factoring the quadratic. Common mistakes include:

• Forgetting to multiply both sides by 2, leading to working with \(\mathrm{b^2 + 4b - 126 = 0}\)
• Factoring errors that lead to incorrect values for b

These calculation mistakes can produce various incorrect intermediate values, causing confusion and leading to guessing among the remaining choices.

The Bottom Line:

This problem requires careful attention to translating English relationships into correct mathematical expressions, followed by systematic algebraic manipulation. The key insight is recognizing that "A is 4 greater than B" means \(\mathrm{A = B + 4}\), not \(\mathrm{B = A + 4}\).