A water level is maintained up to a fixed height H in a cylindrical...

Let the hole be made at a height $y$ from the bottom of the vessel.
The depth of the hole from the free surface of the water is $H - y$.

By Torricelli's theorem, the velocity of efflux is:
$v = \sqrt{2g(H - y)}$

The time taken by the water stream to reach the horizontal plane is:
$t = \sqrt{\dfrac{2y}{g}}$

The horizontal range $x$ of the water stream is given by:
$x = vt$

Substituting the expressions for $v$ and $t$, we get:
$x = \sqrt{2g(H - y)} \times \sqrt{\dfrac{2y}{g}} = 2\sqrt{y(H - y)}$

To maximize the horizontal range $x$, the quantity inside the square root, i.e., $y(H - y)$, must be maximum.
Let $f(y) = y(H - y) = Hy - y^2$.

For maximum range, $\dfrac{df(y)}{dy} = 0$.
$\dfrac{d}{dy}(Hy - y^2) = H - 2y = 0$
$\Rightarrow y = \dfrac{H}{2}$

Also, $\dfrac{d^2f(y)}{dy^2} = -2 < 0$, which confirms a maximum.

Therefore, the hole should be made at a height $\dfrac{H}{2}$ from the bottom.