A rectangular prism has edges with lengths of 4 centimeters, 8 centimeters, and 10 centimeters. The space diagonal (the straight-line...

1. TRANSLATE the problem information

• Given information:
  • Rectangular prism with edges: \(\mathrm{4\ cm, 8\ cm, 10\ cm}\)
  • Need space diagonal in form \(\mathrm{2\sqrt{p}}\)
  • Must find the value of p

• What this tells us: We need the 3D distance between opposite vertices, then factor it into the specific form requested.

2. INFER the approach

• The space diagonal connects two opposite vertices through the interior of the prism
• This requires the 3D distance formula: \(\mathrm{D = \sqrt{a^2 + b^2 + c^2}}\)
• After calculating, we'll need to factor the result to match the form \(\mathrm{2\sqrt{p}}\)

3. SIMPLIFY the calculation

• Apply the formula: \(\mathrm{D = \sqrt{4^2 + 8^2 + 10^2}}\)
• Calculate each term:
  \(\mathrm{D = \sqrt{16 + 64 + 100}}\)
  \(\mathrm{D = \sqrt{180}}\)

4. SIMPLIFY to match the required form

• Factor 180 to extract perfect squares: \(\mathrm{180 = 4 \times 45}\)
• Apply square root property:
  \(\mathrm{\sqrt{180} = \sqrt{4 \times 45}}\)
  \(\mathrm{= \sqrt{4} \times \sqrt{45}}\)
  \(\mathrm{= 2\sqrt{45}}\)
• Since \(\mathrm{2\sqrt{45} = 2\sqrt{p}}\), we have \(\mathrm{p = 45}\)

Answer: 45

Why Students Usually Falter on This Problem

Most Common Error Path:

Missing conceptual knowledge: Not knowing the space diagonal formula for rectangular prisms.

Without this formula, students might try to use 2D diagonal formulas or attempt to visualize the problem incorrectly. They may calculate face diagonals instead of the space diagonal, leading to incorrect intermediate values and ultimately wrong answers.

This leads to confusion and guessing.

Second Most Common Error:

Weak SIMPLIFY skill: Making arithmetic errors when computing \(\mathrm{4^2 + 8^2 + 10^2 = 180}\).

Students might calculate \(\mathrm{16 + 64 + 100}\) incorrectly, getting values like 170 or 190. This leads them down the wrong path when factoring, potentially resulting in incorrect values of p.

This may lead them to select incorrect answer choices based on their miscalculated starting point.

The Bottom Line:

This problem requires both knowing the specific 3D distance formula and being careful with arithmetic. The factoring step, while straightforward, depends entirely on having the correct value of 180 to work with.