Question

A cone of radius r centimetres and height h centimetres iss lowered first at a rate of 1 cm/s into a tall cylinder of radius R that is partially filled with water. How fast is the water level rising at the instant the cone is completely submerged?

Answer 1

$z\left( t \right) -coordinate\,\,of\,\,the\,\,cone\,\,vertex\\z\left( 0 \right) =0\\z'\left( t \right) =-1\\y\left( t \right) \,\,-\,\,the\,\,water\,\,level\\y\left( 0 \right) =0\\\pi R^2y\left( t \right) \,\,-\,\,volume\,\,of\,\,the\,\,cylinder\\\frac{1}{3}\pi \left( y\left( t \right) -z\left( t \right) \right) \left( \frac{y\left( t \right) -z\left( t \right)}{h}r \right) ^2-volume\,\,of\,\,the\,\,part\,\,of\,\,cone\,\,in\,\,water\\\pi R^2y\left( t \right) -\frac{\pi r^2}{3h^2}\left( y\left( t \right) -z\left( t \right) \right) ^3=0\\Completely\,\,submerged:\\y\left( t \right) -z\left( t \right) =h\\\pi R^2y'-\frac{\pi r^2}{3h^2}3\left( y'-z' \right) \left( y\left( t \right) -z\left( t \right) \right) ^2=0\\\pi R^2y'-\pi r^2\left( y'+1 \right) =0\Rightarrow y'=\frac{r^2}{R^2-r^2}$