A rectangular vessel when full of water takes 10 minutes to be emptied through an orifice in its bottom. How much time will it take to be emptied when half filled with water

Options

A. 9 minutes
B. 7 minutes
C. 5 minutes
D. 3 minutes

Answer

B

Explanation

Let A and a be cross - sectional areas of the vessel and orifice respectively. Let h be height of water in the vessel at time. Let `(-(dh)/(dt))` represent the rate of fall of level.
Then `A(-(dh)/(dt))=av=a sqrt(2gh)`
`-(dh)/(dt)=(a sqrt(2gh))/(A)`
`-int_(h)^(0)(1)/(sqrt(h))dh=(a sqrt(2g))/(A)int_(0)^(t)dt` or, `-(-2sqrt(h))=(a sqrt(2g)t)/(A)`
`t=(A)/(a)(1)/(sqrt(2g))xx 2sqrt(h)`
`t=(A)/(a)sqrt((2h)/(g))` or, `t prop sqrt(h)`
`therefore (t_(1))/(t_(2))=sqrt((h_(1))/(h_(2)))` or, `t_(2)=t_(1)sqrt((h_(2))/(h_(1)))=10 sqrt((h_(1))/(2h_(1)))`
`= (10)/(sqrt(2))=5sqrt(2)~= 7` min.