Sand is pouring from a pipe at the rate of`12\ c m^3//s` . The falling sand forms a cone on the ground in such a way that the height of the cone is always one-sixth of the radius of the base. How fast is the height of the sand cone increasing when the height is 4 cm?

Explanation

`V=(1)/(3) pi r^(2) h`
`" but "" "h= (r)/(6)`
`rArr " " V=(1)/(3) pi (6h)^(2)h`
`rArr " "V= 12 pi h^(3)`
` (dV)/(dt) =36 pi h^(2). (dh)/(dt)`
when `.(dV)/(dt)=12 cm^(3)//s " ""and"" "h=4 cm`
`(dh)/(dt)= (12)/(36pi (4)^(2)) =(1)/(48pi) cm//sec.`