A rectangular storage container has a volume of 1,680 cubic inches. The container's length x, in inches, is 15 inches...

1. TRANSLATE the problem information

• Given information:
  • \(\mathrm{Volume = 1,680\ cubic\ inches}\)
  • Length x is 15 inches longer than width
  • \(\mathrm{Height = 8\ inches}\)

• What this tells us:
  • \(\mathrm{Length = x\ inches}\)
  • \(\mathrm{Width = (x - 15)\ inches}\)
  • \(\mathrm{Height = 8\ inches}\)

2. INFER the approach needed

• Since we have a rectangular container and know the volume, we need to use the volume formula
• We'll set up an equation using \(\mathrm{V = length \times width \times height}\)
• The goal is to create an equation that matches one of the answer choices (all in standard quadratic form)

3. SIMPLIFY by setting up and expanding the volume equation

• \(\mathrm{Volume = length \times width \times height}\)
• \(\mathrm{1,680 = x \times (x - 15) \times 8}\)
• \(\mathrm{1,680 = 8x(x - 15)}\)
• \(\mathrm{1,680 = 8x^2 - 120x}\)

4. SIMPLIFY further by isolating terms

• Divide both sides by 8: \(\mathrm{210 = x^2 - 15x}\)
• Rearrange to standard form: \(\mathrm{0 = x^2 - 15x - 210}\)

Answer: A

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students incorrectly interpret "length is 15 inches longer than width" and set up \(\mathrm{width = (x + 15)}\) instead of \(\mathrm{width = (x - 15)}\).

If length is x and they think width is \(\mathrm{(x + 15)}\), their equation becomes:

\(\mathrm{1,680 = x(x + 15)(8) = 8x^2 + 120x}\)

This leads to: \(\mathrm{x^2 + 15x = 210}\), or \(\mathrm{0 = x^2 + 15x - 210}\)

This may lead them to select Choice C \(\mathrm{(0 = x^2 + 15x - 210)}\)

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly set up the initial equation but make sign errors when rearranging to standard form.

They might move terms incorrectly and get: \(\mathrm{x^2 - 15x + 210 = 0}\)

This may lead them to select Choice B \(\mathrm{(0 = x^2 - 15x + 210)}\)

The Bottom Line:

This problem tests whether students can correctly translate relational language ("15 inches longer than") into algebraic expressions and then manipulate the resulting equation without computational errors.