An ice-cream shop uses a scoop that forms a hemisphere of ice cream with radius 3 centimeters. One liter is...

1. TRANSLATE the problem information

• Given information:
  • Each scoop forms a hemisphere with \(\mathrm{radius = 3\,cm}\)
  • Container capacity = \(\mathrm{1\,liter = 1{,}000\,cm^3}\)
  • Need to find: number of scoops to fill container

• What this tells us: We need to compare the volume of one scoop to the total volume needed

2. INFER the solution approach

• To find how many scoops fit, we need:
  • Volume of one scoop (hemisphere)
  • Total volume to fill \(\mathrm{(1{,}000\,cm^3)}\)
  • Then divide: total volume ÷ volume per scoop

• Start by finding the volume of one hemisphere

3. SIMPLIFY the hemisphere volume calculation

• Volume of hemisphere = \(\mathrm{\frac{2}{3}\pi r^3}\)
• With \(\mathrm{r = 3\,cm}\): \(\mathrm{V = \frac{2}{3}\pi(3^3)}\)

\(\mathrm{V = \frac{2}{3}\pi(27)}\)

\(\mathrm{V = 18\pi\,cm^3}\)

• Calculate numerical value: \(\mathrm{18\pi \approx 18 \times 3.14159 \approx 56.55\,cm^3}\) (use calculator)

4. SIMPLIFY to find the number of scoops

• Number of scoops = \(\mathrm{1{,}000 \div 56.55 \approx 17.67}\) (use calculator)
• Round to nearest whole number: 18 scoops

Answer: (C) 18

Why Students Usually Falter on This Problem

Most Common Error Path:

Conceptual confusion about hemisphere vs sphere: Students may use the full sphere volume formula \(\mathrm{V = \frac{4}{3}\pi r^3}\) instead of the hemisphere formula \(\mathrm{V = \frac{2}{3}\pi r^3}\).

Using sphere volume:

\(\mathrm{V = \frac{4}{3}\pi(27)}\)

\(\mathrm{V = 36\pi \approx 113.1\,cm^3}\)

This gives: \(\mathrm{1000 \div 113.1 \approx 8.84 \approx 9\,scoops}\)

This may lead them to select Choice (A) (10) as the closest option, or cause confusion since 9 isn't available.

Second Most Common Error:

Poor SIMPLIFY execution: Students make calculation errors when approximating π or performing the final division, leading to significantly different results.

For example, using \(\mathrm{\pi \approx 3}\) gives:

\(\mathrm{18 \times 3 = 54}\)

\(\mathrm{1000 \div 54 \approx 18.5}\)

still giving 18 or 19 scoops. But other calculation errors could lead to different answer choices.

The Bottom Line:

This problem tests whether students can distinguish between sphere and hemisphere volumes while managing multi-step calculations involving π approximations and unit conversions.