Question

A conical paper cup is 15 cm high with a radius of 5 cm. Suppose now that the water level is rising at 4 cm/s. How quickly is the water being poured into the cup when the water is 8 cm deep?

Answer 1

\[suppose~the~ height ~of~water~be~h~at~any~time~t~and~radius~of~water~column~be~r\]
then volume of water at that time is
\[V=\1/3\pi r^2 h=\pi/3 \left( \frac{ h }{ 3 } \right)^2h=\frac{ 1 }{ 27 }\pi h^3\]
\[\frac{ dV }{ dt }=\frac{ 1 }{ 27 }\pi*3 h^2\frac{ dh }{ dt }=\frac{ \pi }{ 9 }h^2\frac{ dh }{ dt }\]
\[\frac{ dh }{ dt }=4cms/s\]
when h=8cms
\[find~\frac{ dV }{ dt }\]