A container of large uniform cross-sectional area A resting on a horizontal surface, holds two immiscible, non-viscous and incompressible liquids of densities d and 2d, each of height H/2 as shown in the figure. The lower density liquid is open to the atmosphere having pressure `P_0`.

(a) A homogeneous solid cylinder of length `L(LltH//2)`, cross-sectional area `A//5` is immersed such that it floats with its axis vertical at the liquid-liquid interface with length `L//4` in the denser liquid. Determine:
(i) the density D of the solid and
(ii) the total pressure at the bottom of the container.
(b) The cylinder is removed and the original arrangement is restored. A tiny hole of area `s(slt ltA)` is punched on the vertical side of the container at a height `h(hltH//2)`. Determine:
(i) the initial speed of efflux of the liquid at the hole,
(ii) the horizontal distance x travelled by the liquid initially, and
(iii) the height `h_m` at which the hole should be punched so that the liquid travels the maximum distance `x_m` initially. Also calculate `x_m`.