A) \[162\,\,c{{m}^{3}}/s\]
B) \[172\,\,c{{m}^{3}}/s\]
C) \[182\,\,c{{m}^{3}}/s\]
D) \[192\,\,c{{m}^{3}}/s\]
Correct Answer: D
Solution :
[d] |
Let A be the vertex and AO the axis of the cone. |
Let \[O'A=h\] be the depth of water in the cone. |
In \[\Delta AO'C,\] |
\[\tan 30{}^\circ =\frac{O'C}{h}\] or \[O'C=\frac{h}{\sqrt{3}}=radius\] |
\[V=\] Volume of water in the cone |
\[=\frac{1}{3}\pi {{(O'C)}^{2}}\times AO'\] |
\[=\frac{1}{3}\pi \left( \frac{{{h}^{2}}}{3} \right)\times h=\frac{\pi }{9}{{h}^{3}}\] |
or \[\frac{dV}{dt}=\frac{\pi }{3}{{h}^{2}}\frac{dh}{dt}\] ? (1) |
But given that depth of water increases at the rate of 1 cm/s |
\[\frac{dh}{dt}=1\,\,cm/s\] ?. (2) |
From (1) and (2), \[\frac{dV}{dt}=\frac{\pi {{h}^{2}}}{3}\] |
When \[h=24\,\,cm,\] the rate of increase of volume is |
\[\frac{dV}{dt}=\frac{\pi {{(24)}^{2}}}{3}=192\,\,c{{m}^{3}}/s\]. |
You need to login to perform this action.
You will be redirected in
3 sec