Question

The pressure at half the depth of a lake is equal to two-third pressure at the bottom of the lake. So the depth 'h' of the lake is (\(ρ\) = density of water in the lake, \(g\) = acceleration due to gravity, \(P_0\) = atmospheric pressure).

• A. \(\frac{2P_0}{ρg}\)
• B. \(\frac{P_0}{ρg}\)
• C. \(\frac{2P_0}{3ρg}\)
• D. \(\frac{P_0}{2ρg}\)

Hint:

Write absolute pressure at half depth and at the bottom.

Answer 1

The Correct Option is A

Step 1: Clear the fraction:
$3\left(P_0 + \frac{\rho g h}{2}\right) = 2\left(P_0 + \rho g h\right)$.

Step 2: Simplify:
$3P_0 + 1.5\rho g h = 2P_0 + 2\rho g h$, so $P_0 = 0.5\rho g h$ and $h = \frac{2P_0}{\rho g}$.

Final Answer:
The depth is $\frac{2P_0}{\rho g}$, option (A). \[ \boxed{\frac{2P_0}{\rho g}} \]