question_answer

A spherical iron ball of 10 cm radius is coated with a layer of ice of uniform thickness that melts at a rate of \[50\,\,c{{m}^{3}}/\min .\]When the thickness of ice is 5 cm, then the rate (in cm/min.) at which of the thickness of ice decreases, is                                                                               [JEE MAIN Held on 09-01-2020 Morning]

A) \[\frac{1}{36\pi }\]        

B) \[\frac{1}{18\pi }\]

C) \[\frac{1}{54\pi }\]        

D) \[\frac{5}{6\pi }\]

Correct Answer: B

Solution :

[b] \[V=\frac{4}{3}\pi \left[ {{\left( 10+\omega  \right)}^{3}}-{{10}^{3}} \right]\] \[\frac{dv}{dt}=\frac{4}{3}\pi \left[ 3{{\left( 10+\omega  \right)}^{2}}\frac{d\omega }{dt} \right]=50\] when \[\omega =5\] \[\Rightarrow \,\,\,\,\,\,\,\frac{d\omega }{dt}=\frac{3\times 50}{4\pi {{.3.15}^{2}}}\] \[=\frac{1}{18\pi }\]