QJEEM22Ma5539: JEE Main 2022 Mathematics - Class 12 JEEM22Ma5539 Solution: Water is being filled at the rate of 1 cm 3 / sec in a right…

$\because$ $V = {1 \over 3}\pi {r^2}h$ and ${r \over h} = {7 \over {35}} = {1 \over 5}$

$\Rightarrow V = {1 \over {75}}\pi {h^3}$

${{dV} \over {dt}} = {1 \over {25}}\pi {h^2}{{dh} \over {dt}} = 1$

$\Rightarrow {{dh} \over {dt}} = {{25} \over {\pi {h^2}}}$

Now, $S = \pi rl = \pi \left( {{h \over 5}} \right)\sqrt {{h^2} + {{{h^2}} \over {25}}} = {\pi \over {25}}\sqrt {26} {h^2}$

$\Rightarrow {{dS} \over {dt}} = {{2\sqrt {26} \pi h} \over {25}}\,.\,{{dh} \over {dt}} = {{2\sqrt {26} } \over h}$

$\Rightarrow {{dS} \over {d{t_{(h = 10)}}}} = {{\sqrt {26} } \over 5}$