A draining conical reservoir. Water is flowing at the rate of `50m^(3)//"min"` from a shallow concrete conical reservoir (vertex down) of base radius 45 m and height 6 m.
(a) How fast is the water level falling when the water is 5 m deep.
(b) How fast is the radius of the water's surfce changing then ? Answer in cm/min.

Explanation

`(h)/(r)=(6)/(45)=(2)/(15)`
`rArr " " (dh)/(dt)=(2)/(15) (dr)/(dt)`
`h=5m rArr "" r=(15xx5)/(2)=(75)/(2)`
`(dV)/(dt)=(pi)/(3)[2rh(dr)/(dt)+r^(2)(dh)/(dt)]`
`rArr" " (150)/(pi)=75xx5 xx(dr)/(dt)((75)/(2))^(2)xx(2)/(15)(dr)/(dt)`
`=(75xx5+(75xx75xx2)/(4xx15))(dr)/(dt)`
`rArr " " (dr)/(dt)=(2xx150)/(3xx5xx75pi)=(4)/(15pi)m//"min"`
`=(400)/(15pi)cm//"min"`
`rArr " " (dh)/(dt)=(2)/(15)xx(400)/(15pi)=(800)/(225pi)cm//"min"`