A cylindrical water tank has a circular base. The distance around the base (the circumference) is 16pi meters, and the...

1. TRANSLATE the problem information

• Given information:
  • Circumference of base: \(\mathrm{C = 16π}\) meters
  • Water depth when full: \(\mathrm{h = 9}\) meters
  • Need to find: Volume in cubic meters

2. INFER the approach needed

• To find cylinder volume, we need the radius and height
• We have the height (9 meters) but need to find radius from circumference
• Strategy: Use circumference formula to find radius, then apply volume formula

3. SIMPLIFY to find the radius

• Start with circumference formula: \(\mathrm{C = 2πr}\)
• Substitute known values: \(\mathrm{16π = 2πr}\)
• Solve for r: \(\mathrm{r = 16π ÷ (2π) = 8}\) meters

4. SIMPLIFY to calculate the volume

• Use cylinder volume formula: \(\mathrm{V = πr²h}\)
• Substitute our values: \(\mathrm{V = π(8)²(9)}\)
• Calculate: \(\mathrm{V = π(64)(9) = 576π}\) cubic meters

Answer: (D) \(\mathrm{576π}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students misread "distance around the base" and don't recognize this means circumference, or they confuse circumference with radius/diameter.

Instead of setting up \(\mathrm{C = 16π}\), they might think the radius is \(\mathrm{16π}\) directly, leading to \(\mathrm{V = π(16π)²(9)}\), which gives an enormously incorrect result. This leads to confusion and guessing among the available choices.

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly identify they need to find radius from circumference but make algebraic errors.

Common mistake: \(\mathrm{16π = 2πr}\) → \(\mathrm{r = 16π/2 = 8π}\) (forgetting to cancel the π). This gives \(\mathrm{V = π(8π)²(9) = π(64π²)(9) = 576π³}\), which doesn't match any answer choice format. This causes them to get stuck and randomly select an answer.

The Bottom Line:

This problem tests whether students can connect the circumference concept to volume calculation. The key insight is recognizing that "distance around the base" means circumference, then systematically working through the two-step process: circumference → radius → volume.