Water level is maintained upto a fixed height H.The vessel is kept on a horizontal plane.At what height above the bottom should a hole be made in the vessel so that the water stream coming out of the hole strikes the horizontal plane at the greatest distance from the vessel.(Figure in attachment).
Let the desired height be h.
Also, horizontal distance (range) \(x \propto v\sqrt {h}\)
Or, \(x \propto \sqrt {(H-h)h}\)
Let f(h) = (H-h)h
f'(h) = H - 2h =0 for maxima or minima.
\(\Rightarrow h=\frac {H}{2}\) for maxima or minima.
f"(h) = -ve which confirms maxima.
Hence, \(h=\frac {H}{2}\).