A sphere has a radius of 17/5 feet. What is the volume, in cubic feet, of the sphere?

1. TRANSLATE the problem information

• Given information:
  • Sphere has radius = \(\frac{17}{5}\) feet
  • Need to find volume in cubic feet

2. INFER the approach needed

• Since we need sphere volume and have the radius, we should use the sphere volume formula
• The formula is \(\mathrm{V} = \frac{4}{3}\pi\mathrm{r}^3\)

3. SIMPLIFY by substituting and calculating

• Substitute \(\mathrm{r} = \frac{17}{5}\) into \(\mathrm{V} = \frac{4}{3}\pi\mathrm{r}^3\):

\(\mathrm{V} = \frac{4}{3}\pi\left(\frac{17}{5}\right)^3\)

• Calculate the cube of the fraction:

\(\left(\frac{17}{5}\right)^3 = \frac{17^3}{5^3} = \frac{4,913}{125}\) (use calculator for \(17^3\))

• Multiply by the coefficient:

\(\mathrm{V} = \frac{4}{3}\pi\left(\frac{4,913}{125}\right)\)
\(\mathrm{V} = \frac{4 \times 4,913}{3 \times 125} \times \pi\)
\(\mathrm{V} = \frac{19,652}{375} \times \pi\) (use calculator for \(4 \times 4,913\))
\(\mathrm{V} = \frac{19,652\pi}{375}\)

Answer: D. \(\frac{19,652\pi}{375}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students make computational errors when calculating \(\left(\frac{17}{5}\right)^3\) or in the subsequent fraction arithmetic. They might incorrectly compute \(17^3\), or make mistakes when multiplying fractions like \(\frac{4}{3} \times \frac{4,913}{125}\). These calculation errors can lead to various incorrect numerical results, causing them to select wrong answer choices like Choice A (\(\frac{5\pi}{17}\)) or Choice C (\(\frac{32\pi}{5}\)).

Second Most Common Error:

Conceptual confusion about the radius value: According to the provided solution, some students might misinterpret the radius as \(\sqrt{\frac{17}{5}}\) instead of \(\frac{17}{5}\). This fundamental misreading of the given information leads them through an entirely different calculation path, resulting in selection of Choice B (\(\frac{68\pi}{15}\)).

The Bottom Line:

This problem tests both conceptual knowledge of the sphere volume formula and computational precision with fractional arithmetic. Students who know the formula can still fail due to calculation errors, while those who misread the radius value will be led completely astray despite correct formula application.