A solid cylindrical metal rod has a height of 30 centimeters and a radius of 8 centimeters. From this rod,...

1. TRANSLATE the problem setup

• Given information:
  • Cylinder: height = \(30\) cm, radius = \(8\) cm
  • Two identical cones carved out
  • Each cone's base sits on one end of the cylinder
  • The cone vertices meet at the center point of the cylinder

• What this tells us: We need to find cylinder volume minus the volume of two cones

2. INFER the strategy and cone dimensions

• Strategy: Calculate total cylinder volume, then subtract the volume of both carved-out cones

• Key insight: If the cone vertices meet at the center point of the cylinder, then each cone extends from one end to the middle
  • Each cone height = \(30 \div 2 = 15\) cm
  • Each cone radius = \(8\) cm (same as cylinder radius)

3. Calculate cylinder volume

• Using \(\mathrm{V} = \pi\mathrm{r}^2\mathrm{h}\):
  \(\mathrm{V_{cylinder}} = \pi(8)^2(30)\)
  \(= \pi(64)(30)\)
  \(= 1920\pi\) cm³

4. Calculate volume of carved-out cones

• Volume of one cone using \(\mathrm{V} = \frac{1}{3}\pi\mathrm{r}^2\mathrm{h}\):
  \(\mathrm{V_{one\_cone}} = \frac{1}{3}\pi(8)^2(15)\)
  \(= \frac{1}{3}\pi(64)(15)\)
  \(= 320\pi\) cm³

• Total volume of both cones:
  \(\mathrm{V_{two\_cones}} = 2 \times 320\pi\)
  \(= 640\pi\) cm³

5. SIMPLIFY to find remaining volume

• Remaining volume = Original volume - Carved-out volume
  \(\mathrm{V_{remaining}} = 1920\pi - 640\pi\)
  \(= 1280\pi\) cm³

• Convert to decimal (use calculator):
  \(1280 \times 3.14159 \approx 4021.24\) cm³

• Round to nearest cubic centimeter: \(4021\) cm³

Answer: B (4,021)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students misinterpret "vertices meet at center point" and assume each cone has the full cylinder height (\(30\) cm) instead of half the height (\(15\) cm).

Using height = \(30\) cm for each cone:
\(\mathrm{V_{one\_cone}} = \frac{1}{3}\pi(8)^2(30)\)
\(= 640\pi\) cm³
Total cone volume = \(2 \times 640\pi = 1280\pi\) cm³
Remaining volume = \(1920\pi - 1280\pi = 640\pi \approx 2011\) cm³

This leads them to select Choice A (2,011)

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly find that the remaining volume is \(1280\pi\) cm³ but make arithmetic errors when converting to a decimal, either using an incorrect value of π or making calculation mistakes.

This causes confusion and may lead to guessing among the remaining answer choices.

The Bottom Line:

The key challenge is correctly interpreting the geometric constraint that vertices meeting at the center means each cone only extends halfway through the cylinder, not the full height.