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Joint Entrance Examination
Mathematics
Application of Derivatives
Question 83
Question 83 of 212
Medium
MCQ
A spherical iron ball
10
10
10
cm in radius is coated with a layer of ice of uniform thickness that melts at a rate of
50
50
50
cm
3
^3
3
/min. When the thickness of ice is
5
5
5
cm, then the rate at which the thickness of ice decreases is
A
\[{1 \over {36\pi }}\] cm/min
B
\[{1 \over {18\pi }}\] cm/min
C
\[{1 \over {54\pi }}\] cm/min
D
\[{5 \over {6\pi }}\] cm/min
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Official solution
Given that \[{{dv} \over {dt}} = 50\,c{m^3}/\min \] \[ \Rightarrow {d \over {dt}}\left( {{4 \over 3}\pi {r^3}} \right) = 50\] \[ \Rightarrow 4\pi {r^2}{{dr} \over {dt}} = 50\] \[ \Rightarrow {{dr} \over {dt}} = {{50} \over {4\pi {{\left( {15} \right)}^2}}} = {1 \over {18\pi }}\,\,cm/\min \,\,\] (here
r
=
10
+
5
)
r=10+5)
r
=
10
+
5
)
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