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If it's needed, then prove it. I didn't delve into the solution because the dimension of the expression under the root is (kg*m)/((kg/m3)*(m/s2)) = (kg*m)*(m2*s2)/kg = m3*s2. After taking the square root, it will be m3/2*s, and height is measured only in meters to the first power and without any seconds...
Added 1 minute ago
Now I'll also try to understand the solution.
Added 10 minutes ago
From the very beginning, the pressure at height h will not be ρgh, because at the very bottom it will be the largest, and not zero...
Added 1 minute ago
And, if possible, Ment, please provide more details, because the idea of the solution is absolutely correct, but the answer is not yet.
Всё не так плохо как Вы думаете. Всё намного хуже!