If the volume of a frustrum is given as \(V = \frac{\pi h}{3} (R^2 + Rr + r^2)\), find \(\frac{\mathrm d V}{\mathrm d R}\).
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Correct Answer: Option A
Explanation:
\(V = \frac{\pi h}{3} (R^2 + Rr + r^2)\)
\(V = \frac{\pi R^2 h}{3} + \frac{\pi Rr h}{3} + \frac{\pi r^2 h}{3}\)
\(\frac{\mathrm d V}{\mathrm d R} = \frac{2 \pi R h}{3} + \frac{\pi r h}{3}\)
= \(\frac{\pi}{3} (2R + r)\)
\(V = \frac{\pi h}{3} (R^2 + Rr + r^2)\)
\(V = \frac{\pi R^2 h}{3} + \frac{\pi Rr h}{3} + \frac{\pi r^2 h}{3}\)
\(\frac{\mathrm d V}{\mathrm d R} = \frac{2 \pi R h}{3} + \frac{\pi r h}{3}\)
= \(\frac{\pi}{3} (2R + r)\)