If the volume of a frustrum is given as \(V = \frac{\pi h}{3} (R^2 + Rr + r^2)\), find \(\frac{\mathrm d V}{\mathrm d R}\).
A
\(\frac{\pi h}{3} (2R + r)\)
B
\(2R + r + \frac{\pi h}{3}\)
C
\(\frac{\pi h}{3} (2R^2 + r + 2r)\)
D
\(\frac{2R^2}{3} \pi h\)
Verified Correct Answer: A.
Step-by-Step Mathematical Solution
\(V = \frac{\pi h}{3} (R^2 + Rr + r^2)\)
\(V = \frac{\pi R^2 h}{3} + \frac{\pi Rr h}{3} + \frac{\pi r^2 h}{3}\)
\(\frac{\mathrm d V}{\mathrm d R} = \frac{2 \pi R h}{3} + \frac{\pi r h}{3}\)
= \(\frac{\pi}{3} (2R + r)\)
Source: Myschool Past Questions Archive (Record #55369)