A rectangular storage container has dimensions 12 inches by 15 inches by 20 inches. A cylindrical pipe with radius 4...

1. TRANSLATE the problem information

• Given information:
  • Rectangular container: 12 in × 15 in × 20 in
  • Cylindrical pipe: radius = 4 in, runs through the 20-inch length
  • Find: remaining volume to nearest cubic inch
• This tells us we need to find: Container volume - Pipe volume

2. INFER the approach

• The pipe "runs completely through the container along its 20-inch length" means the cylinder extends the full 20 inches
• Strategy: Calculate each volume separately, then subtract

3. Calculate the rectangular container volume

• Volume = length × width × height = \(12 \times 15 \times 20 = 3,600\) cubic inches

4. Calculate the cylindrical pipe volume

• The cylinder has radius 4 inches and height 20 inches (the length it runs through)
• Volume = \(\pi r^2h\)
  \(= \pi(4^2)(20)\)
  \(= \pi(16)(20)\)
  \(= 320\pi\) cubic inches

5. SIMPLIFY the cylindrical volume to a decimal

• \(320\pi \approx 320 \times 3.14159 \approx 1,005.31\) cubic inches (use calculator)

6. Find the remaining volume

• Remaining volume = \(3,600 - 1,005.31 \approx 2,594.69\) cubic inches
• To the nearest cubic inch: 2,595 cubic inches

Answer: (A) 2,595

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students misinterpret which dimension of the container becomes the height of the cylinder. They might use 12 inches or 15 inches instead of recognizing that "along its 20-inch length" means the cylinder height is 20 inches.

If they use height = 12: Volume = \(\pi(4^2)(12) = 192\pi \approx 603\) cubic inches
Remaining: \(3,600 - 603 = 2,997\) cubic inches

This may lead them to select Choice (B) (3,005) as the closest option.

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly set up the problem but make calculation errors with \(\pi\), either using a rough approximation like \(\pi \approx 3\) or making arithmetic errors in the final subtraction.

Using \(\pi \approx 3\): \(320 \times 3 = 960\), so remaining = \(3,600 - 960 = 2,640\)
This doesn't match any answer choice exactly, leading to confusion and guessing.

The Bottom Line:

This problem requires careful reading to understand the spatial relationship between the pipe and container, plus accurate decimal calculations with \(\pi\). Students who rush through the setup or use imprecise \(\pi\) values will struggle to reach the correct answer.