A right circular cylindrical tank has a volume of 72pi cubic feet. The height of the tank is 8 feet....

1. TRANSLATE the problem information

• Given information:
  • Volume of cylindrical tank: \(\mathrm{V = 72\pi}\) cubic feet
  • Height of tank: \(\mathrm{h = 8}\) feet
  • Shape: right circular cylinder
• What we need to find: diameter of the base

2. INFER the approach needed

• To find diameter, we first need the radius (since \(\mathrm{d = 2r}\))
• We have volume and height, so we can use the cylinder volume formula: \(\mathrm{V = \pi r^2h}\)
• Strategy: Use the volume formula to solve for radius, then double it for diameter

3. SIMPLIFY by substituting into the volume formula

• Start with \(\mathrm{V = \pi r^2h}\)
• Substitute known values: \(\mathrm{72\pi = \pi r^2(8)}\)
• Divide both sides by \(\mathrm{8\pi}\): \(\mathrm{\frac{72\pi}{8\pi} = r^2}\)
• This simplifies to: \(\mathrm{9 = r^2}\)

4. SIMPLIFY to find the radius

• Take the square root of both sides: \(\mathrm{r = \sqrt{9}}\)
• Since radius must be positive: \(\mathrm{r = 3}\) feet

5. INFER the final step

• The problem asks for diameter, not radius
• Use the relationship: diameter = \(\mathrm{2 \times radius}\)
• Calculate: \(\mathrm{d = 2(3) = 6}\) feet

Answer: B) 6 feet

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students make algebraic mistakes when solving the equation \(\mathrm{72\pi = \pi r^2(8)}\). They might incorrectly divide by 8 instead of \(\mathrm{8\pi}\), getting \(\mathrm{9\pi = r^2}\), which leads to \(\mathrm{r = 3\sqrt{\pi} \approx 5.3}\) feet, and then \(\mathrm{d \approx 10.6}\) feet. This doesn't match any answer choice exactly, leading to confusion and guessing.

Second Most Common Error:

Poor INFER reasoning: Students find the radius correctly (\(\mathrm{r = 3}\)) but forget that the problem asks for diameter, not radius. They stop at finding \(\mathrm{r = 3}\) and select Choice A (3 feet), missing the final step of doubling the radius.

The Bottom Line:

This problem tests whether students can work systematically through a multi-step process: use the correct formula, solve algebraically, and remember to answer the actual question being asked (diameter, not radius).