Sand is being poured onto a conical pile at the constant rate of  such that the height of the cone is always one half of the radius of its base. How fast is the height of the pile increasing when the sand is 5 cm deep.

Explanation

`V=frac{1}{3} pi r^{2} h`


`Rightarrow V=frac{1}{3} pi(2 h)^{2} h[because h=frac{r}{2}]`


`Rightarrow V=frac{4}{3} pi h^{3}`


`Rightarrow frac{d V}{d t}=4 pi h^{2} frac{d h}{d t}`


`Rightarrow 50=4 pi h^{2} frac{d h}{d t}`


`Rightarrow frac{d h}{d t}=frac{50}{4 pi(5)^{2}}`


`Rightarrow frac{d h}{d t}=frac{1}{2 pi} {cm} / {min}`