A rectangular storage container has a total surface area of 254 square inches. The length of the container is 9...

1. TRANSLATE the problem information

• Given information:
  • Surface area = 254 square inches
  • Length = 9 inches
  • Width = 7 inches
  • Find: height

• What this tells us: We need to use the surface area formula to find the missing dimension.

2. INFER the correct approach

• Since we have a rectangular container with known surface area, we need the surface area formula: \(\mathrm{SA = 2(lw + lh + wh)}\)
• Strategy: Substitute known values and solve for height (h)

3. TRANSLATE into mathematical equation

Set up the equation:

\(\mathrm{254 = 2(9×7 + 9h + 7h)}\)

4. SIMPLIFY step by step

• First, calculate the known area: \(\mathrm{9×7 = 63}\)
• Combine like terms with h: \(\mathrm{9h + 7h = 16h}\)
• Equation becomes: \(\mathrm{254 = 2(63 + 16h)}\)
• Distribute the 2: \(\mathrm{254 = 126 + 32h}\)
• Subtract 126 from both sides: \(\mathrm{128 = 32h}\)
• Divide by 32: \(\mathrm{h = 4}\)

Answer: B) 4

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students may confuse surface area with volume and use \(\mathrm{V = lwh}\) instead of the surface area formula.

Using volume formula: \(\mathrm{254 = 9×7×h = 63h}\), so \(\mathrm{h = 254/63 ≈ 4.03}\), leading them to round to 4. While this accidentally gives the right answer choice, it's based on wrong reasoning and wouldn't work for other problems.

Second Most Common Error:

Poor SIMPLIFY execution: Students make algebraic errors, particularly when distributing the 2 or combining like terms.

For example, incorrectly getting \(\mathrm{254 = 2(63) + 16h}\) instead of \(\mathrm{254 = 2(63 + 16h)}\), leading to \(\mathrm{254 = 126 + 16h}\), then \(\mathrm{128 = 16h}\), so \(\mathrm{h = 8}\). This leads them to select Choice D (8).

The Bottom Line:

This problem requires careful attention to which formula applies (surface area vs volume) and methodical algebraic manipulation. The key insight is recognizing that a rectangular container's surface area includes all six faces.