The area A of a trapezoid is given by the formula \(\mathrm{A = \frac{1}{2}h(b_1 + b_2)}\), where h is the...

1. TRANSLATE the problem information

• Given information:
  • Area = 20
  • Longer base = 8
  • Height = shorter base (this is the key relationship!)
  • Need to find: height
• What this tells us: We can use one variable (h) for both height and shorter base

2. INFER the solution approach

• Since we have the area formula and know relationships between variables, we can set up an equation
• Let \(\mathrm{h = height = shorter\ base\ (b_1)}\), and \(\mathrm{b_2 = 8}\)

3. TRANSLATE into the area formula

Using \(\mathrm{A = \frac{1}{2}h(b_1 + b_2)}\):

\(\mathrm{20 = \frac{1}{2}h(h + 8)}\)

4. SIMPLIFY to get a quadratic equation

• Multiply both sides by 2: \(\mathrm{40 = h(h + 8)}\)
• Distribute: \(\mathrm{40 = h^2 + 8h}\)
• Rearrange: \(\mathrm{h^2 + 8h - 40 = 0}\)

5. INFER that we need the quadratic formula

• This quadratic doesn't factor easily, so use the quadratic formula
• Where \(\mathrm{a = 1}\), \(\mathrm{b = 8}\), \(\mathrm{c = -40}\)

6. SIMPLIFY using the quadratic formula

\(\mathrm{h = \frac{-8 ± \sqrt{8^2 - 4(1)(-40)}}{2(1)}}\)

\(\mathrm{h = \frac{-8 ± \sqrt{64 + 160}}{2}}\)

\(\mathrm{h = \frac{-8 ± \sqrt{224}}{2}}\)

7. SIMPLIFY the radical

• Find perfect square factors of 224: \(\mathrm{224 = 16 × 14 = 4^2 × 14}\)
• So \(\mathrm{\sqrt{224} = 4\sqrt{14}}\)
• Therefore: \(\mathrm{h = \frac{-8 ± 4\sqrt{14}}{2} = -4 ± 2\sqrt{14}}\)

8. APPLY CONSTRAINTS to select the valid solution

• Height must be positive in the real world
• Check: Since \(\mathrm{\sqrt{14} \gt 3}\), we have \(\mathrm{2\sqrt{14} \gt 6}\), so \(\mathrm{-4 + 2\sqrt{14} \gt 2}\) (positive ✓)
• Check: \(\mathrm{-4 - 2\sqrt{14} \lt -10}\) (negative ✗)

Answer: B) \(\mathrm{-4 + 2\sqrt{14}}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students make algebraic errors when expanding \(\mathrm{h(h + 8) = 40}\) or applying the quadratic formula, particularly with the signs and arithmetic in the discriminant calculation.

For example, they might calculate the discriminant as \(\mathrm{64 - 160 = -96}\) instead of \(\mathrm{64 + 160 = 224}\), leading them to think there's no real solution. Or they might incorrectly simplify \(\mathrm{\sqrt{224}}\), getting stuck with an unsimplified radical. This leads to confusion and potentially guessing among the answer choices.

Second Most Common Error:

Inadequate APPLY CONSTRAINTS reasoning: Students correctly solve the quadratic but select the negative solution \(\mathrm{-4 - 2\sqrt{14}}\), not recognizing that height must be positive in a geometric context.

Since none of the answer choices are obviously negative (they all involve \(\mathrm{\sqrt{14}}\)), students might not immediately recognize that \(\mathrm{-4 - 2\sqrt{14}}\) is negative, potentially leading them to select Choice A (\(\mathrm{4 - 2\sqrt{14}}\)) if they made a sign error, or get confused between the positive and negative forms.

The Bottom Line:

This problem combines several challenging elements: translating a complex word relationship into algebra, solving a non-obvious quadratic equation, simplifying radicals, and applying real-world constraints. Students need strong algebraic skills and the insight to recognize which mathematical solution makes physical sense.