A company manufactures cardboard boxes that are right rectangular prisms. A standard box design has a length of 20 inches...

1. TRANSLATE the problem information

• Given information:
  • Length = 20 inches
  • Width = 12 inches
  • Total surface area = 1120 square inches
• Need to find: height of the box

2. TRANSLATE the surface area relationship

• For a rectangular prism: \(\mathrm{SA = 2(lw + lh + wh)}\)
• This accounts for all 6 faces: 2 faces each of lw, lh, and wh

3. SIMPLIFY by substituting known values

• \(\mathrm{1120 = 2(20 \times 12 + 20h + 12h)}\)
• Calculate the known area: \(\mathrm{20 \times 12 = 240}\)
• Combine like terms with h: \(\mathrm{20h + 12h = 32h}\)
• So: \(\mathrm{1120 = 2(240 + 32h)}\)

4. SIMPLIFY to solve for h

• Distribute the 2: \(\mathrm{1120 = 480 + 64h}\)
• Subtract 480 from both sides: \(\mathrm{640 = 64h}\)
• Divide both sides by 64: \(\mathrm{h = 10}\)

Answer: B) 10

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students may forget that surface area includes ALL faces of the rectangular prism, not just the unique faces. They might use \(\mathrm{SA = lw + lh + wh}\) (missing the factor of 2), leading to:

\(\mathrm{1120 = 240 + 32h}\)
\(\mathrm{880 = 32h}\)
\(\mathrm{h = 27.5}\)

Since 27.5 isn't among the choices, this leads to confusion and guessing.

Second Most Common Error:

Poor SIMPLIFY execution: Students make arithmetic errors when combining terms or performing division. A common mistake is calculating \(\mathrm{640 \div 64}\) incorrectly, perhaps getting 8 instead of 10.

This may lead them to select Choice A) 8.

The Bottom Line:

This problem tests whether students remember that rectangular prisms have 6 faces (not 3) and can handle multi-step algebraic manipulation without arithmetic errors. The key insight is recognizing that the "2" in the surface area formula is crucial.