A hemisphere is half of a sphere. If a hemisphere has a radius of 27 inches, which of the following...

1. TRANSLATE the problem information

• Given information:
  • Hemisphere with radius = 27 inches
  • Need to find volume in cubic inches

2. INFER the correct volume formula

• Since we know the sphere volume formula \(\mathrm{V} = \frac{4}{3}\pi\mathrm{r}^3\)
• A hemisphere is exactly half of a sphere
• Therefore: Hemisphere volume = \(\frac{1}{2} \times \frac{4}{3}\pi\mathrm{r}^3 = \frac{2}{3}\pi\mathrm{r}^3\)

3. SIMPLIFY by substituting the given radius

• \(\mathrm{V} = \frac{2}{3}\pi(27)^3\)
• First calculate \(27^3 = 19,683\) (use calculator)
• \(\mathrm{V} = \frac{2}{3}\pi(19,683)\)

4. SIMPLIFY the final calculation

• \(\mathrm{V} = \frac{2}{3} \times \pi \times 19,683\)
• \(\mathrm{V} \approx \frac{2}{3} \times 3.14159 \times 19,683\) (use calculator)
• \(\mathrm{V} \approx 41,223\) cubic inches

5. APPLY CONSTRAINTS to select closest answer

• Looking at the choices: A. 1,500, B. 6,100, C. 30,900, D. 41,200
• Our result 41,223 is closest to 41,200

Answer: D. 41,200

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students use the full sphere volume formula instead of recognizing they need half of it.

They calculate \(\mathrm{V} = \frac{4}{3}\pi(27)^3 \approx 82,446\) cubic inches, which doesn't match any answer choice exactly. This leads to confusion and guessing, or they might incorrectly select the closest choice if available.

Second Most Common Error:

Poor SIMPLIFY execution: Students make calculation errors with \(27^3\) or the final multiplication.

For example, they might miscalculate \(27^3\) as something like 2,187 (which is \(3^7\)) instead of 19,683, leading them to get \(\mathrm{V} \approx 4,580\), which might lead them to select Choice B (6,100) as the closest.

The Bottom Line:

This problem tests whether students can connect the relationship between spheres and hemispheres conceptually, then execute multi-step calculations accurately. The key insight is recognizing that "half of a sphere" means you take half the volume, not just apply a different formula entirely.