Two solid right circular cylinders, P and Q, have the same height of 48 centimeters. The diameter of cylinder Q...

1. TRANSLATE the problem information

• Given information:
  • Both cylinders have height \(\mathrm{h = 48\ cm}\)
  • Diameter of Q is 25% greater than diameter of P
  • Volume of P is \(\mathrm{32{,}000\pi}\) cubic cm
  • Need to find volume of Q

2. INFER the most efficient approach

• Since both cylinders have the same height, we can use proportional reasoning
• When cylinders have equal heights, their volumes are proportional to the squares of their radii (or diameters)
• This means we don't need to find the actual radius values - we can work with scaling factors

3. TRANSLATE the diameter relationship

• "25% greater" means: \(\mathrm{d_Q = d_P + 0.25d_P = 1.25d_P}\)
• Since radius is half the diameter: \(\mathrm{r_Q = 1.25r_P}\)

4. INFER the volume scaling relationship

• Volume formula: \(\mathrm{V = \pi r^2h}\)
• Since heights are equal: \(\mathrm{\frac{V_Q}{V_P} = \frac{(r_Q)^2}{(r_P)^2} = \left(\frac{r_Q}{r_P}\right)^2}\)
• Substituting our scaling factor: \(\mathrm{\frac{V_Q}{V_P} = (1.25)^2 = 1.5625}\)

5. SIMPLIFY the scaling calculation

• \(\mathrm{(1.25)^2 = 1.5625 = \frac{25}{16}}\) (use calculator if needed)
• Therefore: \(\mathrm{V_Q = V_P \times \frac{25}{16}}\)

6. SIMPLIFY to find the final answer

• \(\mathrm{V_Q = 32{,}000\pi \times \frac{25}{16}}\)
• \(\mathrm{V_Q = 32{,}000\pi \times 25 \div 16}\)
  \(\mathrm{= 800{,}000\pi \div 16}\)
  \(\mathrm{= 50{,}000\pi}\)

Answer: D) \(\mathrm{50{,}000\pi}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students try to find the actual radius of cylinder P first, then calculate the radius of Q, then find its volume. While this approach works, it's unnecessarily complicated and creates more opportunities for calculation errors.

They might solve: \(\mathrm{32{,}000\pi = \pi r^2(48)}\), so \(\mathrm{r^2 = \frac{32{,}000}{48} = \frac{2000}{3}}\), then \(\mathrm{r = \sqrt{\frac{2000}{3}}}\), then calculate \(\mathrm{r_Q = 1.25\sqrt{\frac{2000}{3}}}\), and finally \(\mathrm{V_Q = \pi\left[1.25\sqrt{\frac{2000}{3}}\right]^2(48)}\). This lengthy path often leads to arithmetic mistakes and may cause them to select an incorrect answer or abandon the problem.

Second Most Common Error:

Poor TRANSLATE reasoning: Students misinterpret "25% greater" as meaning Q's diameter is 0.25 times P's diameter instead of 1.25 times P's diameter.

Using the scaling factor of 0.25 instead of 1.25, they calculate \(\mathrm{V_Q = 32{,}000\pi \times (0.25)^2}\)
\(\mathrm{= 32{,}000\pi \times 0.0625}\)
\(\mathrm{= 2{,}000\pi}\). Since this value doesn't match any answer choice, this leads to confusion and guessing.

The Bottom Line:

This problem rewards students who recognize that proportional relationships can eliminate intermediate calculations. The key insight is that equal heights allow direct volume scaling without finding individual dimensions.