A rectangular prism has a length of 10 inches and a width of 5 inches. The total surface area of...

1. TRANSLATE the problem information

• Given information:
  • Length = 10 inches
  • Width = 5 inches
  • Total surface area = 400 square inches
  • Need to find: height

• What this tells us: We have a 3D geometry problem requiring the surface area formula

2. TRANSLATE the surface area relationship

• Surface area of rectangular prism: \(\mathrm{SA = 2(lw + lh + wh)}\)
• This accounts for all 6 faces: 2 bases (lw) + 2 front/back faces (lh) + 2 side faces (wh)

3. SIMPLIFY by substituting known values

• Replace SA = 400, l = 10, w = 5:

\(\mathrm{400 = 2((10)(5) + (10)(h) + (5)(h))}\)

4. SIMPLIFY the expression step-by-step

• Calculate lw: \(\mathrm{400 = 2(50 + 10h + 5h)}\)
• Combine like terms: \(\mathrm{400 = 2(50 + 15h)}\)
• Distribute the 2: \(\mathrm{400 = 100 + 30h}\)

5. SIMPLIFY to isolate the variable

• Subtract 100 from both sides: \(\mathrm{300 = 30h}\)
• Divide by 30: \(\mathrm{h = 10}\)

Answer: B (10 inches)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students make arithmetic errors when combining the height terms (10h + 5h) or when distributing the 2.

Common mistakes include getting \(\mathrm{2(50 + 25h)}\) instead of \(\mathrm{2(50 + 15h)}\), leading to \(\mathrm{400 = 100 + 50h}\), which gives \(\mathrm{h = 6}\). Since 6 isn't an answer choice, this leads to confusion and guessing.

Second Most Common Error:

Incomplete TRANSLATE reasoning: Students forget that surface area includes ALL faces of the rectangular prism and try to use area formulas for individual faces instead of the complete surface area formula.

They might calculate \(\mathrm{2(lw) = 2(50) = 100}\) and then try to work with the remaining 300 square inches incorrectly. This may lead them to select Choice C (15) by guessing from the larger values.

The Bottom Line:

This problem requires careful algebraic manipulation with multiple terms. Students must stay organized through each step and remember that rectangular prisms have 6 faces, not just the obvious front and back.