Question

A cone of radius 𝑟 centimeters and height ℎ centimeters is lowered point first at

a rate of 1 cm/s into a tall cylinder of radius 𝑅 centimeters that is partially filled with

water. How fast is the water level rising at the instant the cone is completely

submerged?

Answer 1

$V-volume\,\,of\,\,cone\,\,in\,\,water\\y-heigth\,\,of\,\,cone\,\,in\,\,water\\V=\left( \frac{y}{h} \right) ^3\cdot \frac{1}{3}\pi r^2h=\frac{\pi r^2}{3h^2}y^3\\z-heigth\,\,of\,\,water\,\,in\,\,cylinder\,\,above\,\,the\,\,initial\,\,level\\z=\frac{V}{\pi R^2}=\frac{\frac{\pi r^2}{3h^2}y^3}{\pi R^2}=\frac{r^2}{3h^2R^2}y^3\\\frac{dz}{dt}=\frac{r^2}{3h^2R^2}\cdot 3y^2\frac{dy}{dt}=\left[ y=h \right] =\frac{r^2}{R^2}\frac{dy}{dt}=\frac{r^2}{R^2}$