The density P of a spherical ball of diameter d and mass m is given by

Analyze the question: We need the density of a sphere in terms of its mass m and its diameter d (not the radius).

Concept: Density = mass ÷ volume, and the volume of a sphere is V = (4/3)πr³.

Solution:

1. Density: ρ = m / V
2. Volume of a sphere: V = (4/3)πr³
3. The diameter is d, so r = d/2. Substitute:

V = (4/3)π(d/2)³ = (4/3)π(d³/8) = πd³/6

4. Therefore:

ρ = m ÷ (πd³/6) = 6m / (πd³)

Checking the options:

• A: 3m/(4πd³) is what you get if you wrongly use r = d instead of r = d/2.
• B: 4πmd³ is dimensionally wrong, since mass would be in the numerator along with d³.
• C: 6m/(πd³) matches our result.
• D: 3m/(2πd³) does not match.

Answer: C

Tip: Whenever a question gives the diameter, replace r with d/2 right away. The volume of a sphere in terms of diameter is πd³/6, which is worth memorizing. Option A is the trap for students who forget to convert the diameter to the radius.