A) \[\frac{1}{2}S\frac{dr}{dt}\]
B) \[\frac{1}{2}r\frac{dS}{dt}\]
C) \[r\frac{dS}{dt}\]
D) \[\frac{1}{2}{{r}^{2}}\frac{dS}{dt}\]
Correct Answer: B
Solution :
[b] Surface area of sphere \[S=4\pi {{r}^{2}}\] Differentiate both sides w.r.t. t? \[\Rightarrow \frac{dS}{dt}=\frac{8\pi rdr}{dt}\] and Volume \[=V=\frac{4}{3}\pi {{r}^{3}}\] \[\Rightarrow \frac{dV}{dt}=\frac{4}{3}\pi .3{{r}^{2}}\frac{dr}{dt}=4\pi {{r}^{2}}\frac{dr}{dt}\] \[=\frac{4\pi {{r}^{2}}}{8\pi r}.\frac{dS}{dt}=\frac{1}{2}r\frac{dS}{dt}\]
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