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A water tank has the shape of an inverte...

A water tank has the shape of an inverted right circular cone, whose semi-vertical angle is `tan ^(-1)((1)/(2))`. Water is poured into it at a constant rate of 5 cu m/min. Then, the rate (in m/min) at which the level of water is rising at the instant when the depth of water in the tank is 10 m is

A

`(2)/(pi)`

B

`(1)/(15 pi)`

C

`(1)/(5 pi)`

D

`(1)/(10 pi)`

Text Solution

Verified by Experts

The correct Answer is:
C
`theta = tan^(-1) (1)/(2)`
`tan theta = (1)/(2) = (r )/(h) implies h = 2r`
`V = (1)/(3) pi r^(2) h`
`V = (1)/(3) pi (h^(2))/(4). H = (pi h^(3))/(12) , (dv)/(dt) = (pi)/(12) . 3h^(2) (dh)/(dt)`
`implies 5 = (pi)/(4) . 100 xx (dh)/(dt) implies (dh)/(dt) = (1)/(5 pi)`
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