Question

A vessel completely filled with water has holes $A$ and $B$ at depths $h$ and $3h$ from the top respectively. The hole $A$ is a square of side $L$ and $B$ is a circle of radius $r$ . The water flowing out per second from both the holes is same. Then $L$ is equal to

• A. $r^{\frac{1}{2}}\left(\pi \right)^{\frac{1}{2}}\left(3\right)^{\frac{1}{2}}$
• B. $r\left(\pi \right)^{\frac{1}{4}}\left(3\right)^{\frac{1}{4}}$
• C. $r\left(\pi \right)^{\frac{1}{2}}\left(3\right)^{\frac{1}{4}}$
• D. $r^{\frac{1}{2}}\left(\pi \right)^{\frac{1}{3}}\left(3\right)^{\frac{1}{2}}$

Answer 1

Answer: (C)

$\text{A} _{1} \text{V} _{1} = \text{A}_{1} \text{V}_{2}$
$\text{L}^{2} \sqrt{2 \text{gh}} = \pi \text{r}^{2} \sqrt{6 \text{gh}}$
$\text{L}^{4} \text{gh} = \pi ^{2} \text{r}^{4} 3 \text{gh}$
$\text{L} = \left(\pi \right) ^{\frac{1}{2}} \left(\text{r}\right) \left(3\right)^{\frac{1}{4}}$
$\text{L} = \left(\text{r}\right) \left(\pi \right)^{\frac{1}{2}} \left(3\right)^{\frac{1}{4}}$