Question

A growing sand pile . Sand falls from a conveyor belt at the rate of ` 10m^(3)//"min"` on to the top of a conical pile. The height of the pile is always three-eights of the base diameter. How fast are the (a) height and (b) radius changing when the pile is 4 m high ? Answer in cm/min.

Answer 1

`h=(3)/(8)xx(2r)=(3r)/(4)`
`rArr " " (dh)/(dt)=(3)/(4)(dr)/(dt) `

`V=(1)/(3) pi r^(2)h`
`rArr " " (dV)/(dt)=(pi)/(3)[2rh(dr)/(dt)+r^(2)(dh)/(dt)]`
`rArr " " (30)/(pi)=2xx4 xx(16)/(3)(dr)/(dt)+((16)/(3))^(2)(dh)/(dt)`
`rArr " " (30)/(pi)=(128)/(3)xx(4)/(3)(dh)/(dt)+(156)/(9)(dh)/(dt)=((512+256)/(9))(dh)/(dt)`
`rArr " " (dh)/(dt)=(30)/(pi)xx(9)/(256 pi) m//"min"`
` =(9000)/(256pi)cm//"min"`
`rArr " " (dr)/(dt)=(3000)/(256pi)xx(4)/(3)=(3000)/(64pi) cm //"min"`