The formula for the volume, V, of a specialized cone with radius r and a height of 9 is given...

1. TRANSLATE the problem information

• Given information:
  • Volume formula: \(\mathrm{V = 3\pi r^2}\)
  • Volume of the cone: \(\mathrm{75\pi}\)
  • Need to find: radius r

2. TRANSLATE the question into an equation

• Set the given volume equal to the formula:
  \(\mathrm{75\pi = 3\pi r^2}\)

3. SIMPLIFY to isolate r²

• Divide both sides by 3π:
  \(\mathrm{75\pi \div 3\pi = r^2}\)
  \(\mathrm{25 = r^2}\)

4. SIMPLIFY to find r

• Take the square root of both sides:
  \(\mathrm{r = \sqrt{25} = 5}\)

Answer: B. 5

Why Students Usually Falter on This Problem

Most Common Error Path:

Poor SIMPLIFY execution: Students correctly find \(\mathrm{r^2 = 25}\) but forget the final step of taking the square root.

They think "\(\mathrm{r^2 = 25}\)" is the final answer and look for 25 among the choices. Since the question asks for radius r (not r²), they need to take one more step to find \(\mathrm{r = \sqrt{25} = 5}\).

This may lead them to select Choice D (25).

The Bottom Line:

This problem tests whether students can distinguish between r² and r. The algebraic setup is straightforward, but students must complete all steps of the solution process, including the final square root operation.