A sphere has a radius of 15/4 feet. What is the surface area, in square feet, of the sphere?

1. TRANSLATE the problem information

• Given information:
  • Sphere has radius \(\mathrm{r = \frac{15}{4}}\) feet
  • Need to find: surface area in square feet

2. INFER the approach

• For any sphere surface area problem, we need the formula \(\mathrm{A = 4πr^2}\)
• We have the radius, so we can substitute directly into the formula

3. TRANSLATE the setup

• Surface area: \(\mathrm{A = 4πr^2}\)
• Substitute \(\mathrm{r = \frac{15}{4}}\): \(\mathrm{A = 4π(\frac{15}{4})^2}\)

4. SIMPLIFY the calculation

• First, calculate \(\mathrm{(\frac{15}{4})^2}\):
  \(\mathrm{(\frac{15}{4})^2 = \frac{15^2}{4^2} = \frac{225}{16}}\)

• Now substitute: \(\mathrm{A = 4π(\frac{225}{16})}\)

• Multiply: \(\mathrm{A = \frac{4 \times 225π}{16} = \frac{900π}{16}}\)

• Simplify the fraction: \(\mathrm{\frac{900π}{16} = \frac{225π}{4}}\)
  (Divide both 900 and 16 by 4)

Answer: C) \(\mathrm{\frac{225π}{4}}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students correctly set up \(\mathrm{A = 4π(\frac{15}{4})^2}\) but make arithmetic errors when squaring the fraction.

Common mistakes include calculating \(\mathrm{(\frac{15}{4})^2}\) as \(\mathrm{\frac{15}{16}}\) instead of \(\mathrm{\frac{225}{16}}\), or getting confused with fraction arithmetic. This leads to completely wrong intermediate values, and they may select Choice A (\(\mathrm{15π}\)) or Choice B (\(\mathrm{\frac{225π}{16}}\)) depending on where their calculation went wrong.

Second Most Common Error:

Incomplete SIMPLIFY process: Students correctly calculate \(\mathrm{4π(\frac{225}{16}) = \frac{900π}{16}}\) but fail to simplify this final fraction.

They see \(\mathrm{\frac{900π}{16}}\) and think they're done, leading them to select Choice B (\(\mathrm{\frac{225π}{16}}\)) because it looks similar to their unsimplified answer, not realizing they need to reduce the fraction by dividing by 4.

The Bottom Line:

This problem tests careful fraction arithmetic more than conceptual understanding. Students who know the sphere formula usually set up the problem correctly, but success depends entirely on accurate calculation with fractions - both in squaring \(\mathrm{(\frac{15}{4})^2}\) and in simplifying the final result.