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)

${\text{A}}_1{\text{V}}_1={\text{A}}_1{\text{V}}_2$
${\text{L}}^2\sqrt{2\text{gh}}=\pi {\text{r}}^2\sqrt{6\text{gh}}$
${\text{L}}^4\text{gh}={\pi }^2{\text{r}}^{4}3\text{gh}$
$\text{L}={\left(\pi \right)}^{\frac{1}{2}}\left(\text{r}\right){\left(3\right)}^{\frac{1}{4}}$
$\text{L}=\left(\text{r}\right){\left(\pi \right)}^{\frac{1}{2}}{\left(3\right)}^{\frac{1}{4}}$