Jambmaths question:
If the volume of hemisphere is increasing at a steady rate of 18πm3s–1 . At what rate is its radius changing when it is 6m
Option A:
2.50ms–1
Option B:
2.00 ms–1
Option C:
0.25 ms–1
Option D:
0.20 ms–1
Jamb Maths Solution:
$\begin{align} & V=\tfrac{2}{3}\pi {{r}^{3}} \\ & \frac{dV}{dt}=18\pi {{m}^{3}}{{s}^{-1}} \\ & \frac{dV}{dr}=2\pi {{r}^{2}},\text{ }\frac{dr}{dV}=\frac{1}{2\pi {{r}^{2}}} \\ & \frac{dr}{dt}=\frac{dV}{dr}\times \frac{dV}{dt}=\frac{1}{2\pi {{r}^{2}}}\times 18\pi \\ & \frac{dr}{dt}=\frac{9}{{{r}^{2}}} \\ & \text{when }r=6m \\ & \frac{dr}{dt}=\frac{9}{36}m{{s}^{-1}}=0.25m{{s}^{-1}} \\\end{align}$
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