A large open tank has two holes in its wall. Ine is a square hole of side a at a depath of x from the top and the other is a cirular hle of radius r at a depth 4x from the top. When the tank is completely filled with water, the quantities of water flowing out per second from both holes are the same. Then r is equal to

Options

A. `2pi a`
B. `(a)/(2pi)`
C. `(a )/(pi)`
D. `(a)/(sqrt(2pi))`

Answer

D

Explanation

Speed of the water coming out from the hole, `v= sqrt(2gh)`
where h denotes depth of the hole from the free surface of the water in the tank. The quantities of water flowing out per second from both holes are given to be same, therefore `A_(1)v_(1) = A_(2) v_(2)`
where `A_(1) and A_(2)` are the area of cross-sections of hole 1 (i.e. square hole) and hole 2 (i.e, circular hole) and `v_(1) and v_(2)` are speed of water coming out from these holes.
`:. a^(2) sqrt(2gx) = pi r^(2) sqrt(2 g 4x) or a^(2)= 2pi r^(2)`
or `r^(2) = (a^(2))/(2pi) or r = (a)/(sqrt(2pi))`