A storage tank is constructed in the shape of a hemisphere, which is half of a sphere. The volume of...

1. TRANSLATE the problem information

• Given information:
  • Storage tank shaped as hemisphere (half of sphere)
  • Volume = 19,400 cubic feet
  • Need to find radius in feet

• What this tells us: We need the hemisphere volume formula and will work backwards from volume to radius.

2. INFER the approach

• Since hemisphere = half sphere, we'll modify the sphere volume formula
• Strategy: Use \(\mathrm{V_{hemisphere} = \frac{1}{2} \times V_{sphere} = \frac{1}{2} \times \frac{4}{3}\pi r^3 = \frac{2}{3}\pi r^3}\)
• Set this equal to 19,400 and solve for r

3. TRANSLATE the volume relationship into equation

Set up: \(\mathrm{19{,}400 = \frac{2}{3}\pi r^3}\)

4. SIMPLIFY to isolate r³

• Multiply both sides by 3/2:
  \(\mathrm{19{,}400 \times \frac{3}{2} = \pi r^3}\)
  \(\mathrm{29{,}100 = \pi r^3}\)

• Divide by π:
  \(\mathrm{r^3 = \frac{29{,}100}{\pi}}\)

• Using \(\mathrm{\pi \approx 3.14}\) (use calculator):
  \(\mathrm{r^3 \approx 29{,}100 \div 3.14 \approx 9{,}268}\)

5. SIMPLIFY by finding the cube root

• We need to find what number cubed gives approximately 9,268
• Test some values:
  • \(\mathrm{20^3 = 8{,}000}\) (too small)
  • \(\mathrm{21^3 = 9{,}261}\) (very close!)
  • \(\mathrm{22^3 = 10{,}648}\) (too large)

• Since \(\mathrm{21^3 = 9{,}261 \approx 9{,}268}\), our radius is approximately 21 feet

Answer: B. 21

Why Students Usually Falter on This Problem

Most Common Error Path:

Conceptual confusion about hemisphere formula: Many students forget that a hemisphere is exactly half a sphere and incorrectly use the full sphere volume formula \(\mathrm{V = \frac{4}{3}\pi r^3}\) instead of \(\mathrm{V = \frac{2}{3}\pi r^3}\).

When they set up \(\mathrm{19{,}400 = \frac{4}{3}\pi r^3}\), they get:
\(\mathrm{r^3 = \frac{19{,}400 \times 3}{4\pi} \approx \frac{19{,}400 \times 3}{12.56} \approx 4{,}634}\)

Taking the cube root gives approximately \(\mathrm{r \approx 16.7}\), which would lead them to guess or select the closest choice.

This may lead them to select Choice A (11) as the closest "reasonable" answer, or cause confusion and random guessing.

Second Most Common Error:

Weak SIMPLIFY skill in cube root estimation: Students correctly set up the equation and get \(\mathrm{r^3 \approx 9{,}268}\), but struggle with cube root calculations.

Some might incorrectly think that since \(\mathrm{\sqrt{9{,}268} \approx 96}\), then \(\mathrm{\sqrt[3]{9{,}268} \approx 30}\)-ish, leading them to select Choice D (30). Others might get overwhelmed by the calculation and abandon systematic solution.

The Bottom Line:

This problem requires both conceptual clarity about hemisphere geometry and confidence with cube root estimation. Students who rush through the setup or avoid the calculation often miss the systematic approach needed.