A cube has an edge length of 68 inches. A solid sphere with a radius of 34 inches is inside...

1. TRANSLATE the problem information

• Given information:
  • Cube has edge length 68 inches
  • Sphere has radius 34 inches
  • Sphere is inside the cube, touching center of each face
  • Need: volume of space NOT taken up by sphere

• What this tells us: We need to find the difference between the cube's volume and the sphere's volume.

2. INFER the approach

• Since we want the space NOT occupied by the sphere, we calculate: Volume of cube - Volume of sphere
• We'll need both volume formulas, then subtract

3. Calculate the volume of the cube

Using \(\mathrm{V = s^3}\) where s is the edge length:

Volume of cube = \(\mathrm{68^3 = 314,432}\) cubic inches

4. SIMPLIFY the volume of the sphere

Using \(\mathrm{V = \frac{4}{3}\pi r^3}\) where r is the radius:

Volume of sphere = \(\mathrm{\frac{4}{3}\pi(34)^3}\)

First calculate \(\mathrm{34^3 = 39,304}\)

Then: \(\mathrm{\frac{4}{3} \times \pi \times 39,304 = \frac{4}{3} \times 39,304 \times \pi \approx 164,636}\) cubic inches (use calculator)

5. Find the difference

Volume of space = \(\mathrm{314,432 - 164,636 = 149,796}\) cubic inches

Answer: A. 149,796

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students might misinterpret "volume of space not taken up by the sphere" and try to find just the sphere's volume instead of the difference.

This confusion about what the problem is actually asking for leads them to select Choice B (164,500) which is approximately the sphere's volume alone.

Second Most Common Error:

Poor SIMPLIFY execution: Students make calculation errors when computing \(\mathrm{\frac{4}{3}\pi(34)^3}\), particularly getting \(\mathrm{34^3}\) wrong or using an incorrect approximation for \(\mathrm{\pi}\).

Common mistakes include calculating \(\mathrm{34^3}\) as something other than 39,304, or using 3.14 instead of a more accurate \(\mathrm{\pi}\) value. This leads to an incorrect sphere volume and ultimately the wrong final answer.

The Bottom Line:

This problem tests whether students can correctly interpret a "difference" scenario and execute complex volume calculations accurately. The key insight is recognizing that "space not taken up" means subtraction, not just finding one volume.