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(B) Let the depth of the hole below the water surface be $x$. The height of the hole above the ground is $(h - x + H)$.The velocity of efflux is $v =

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$\frac{H+h}{2}$

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(B) Let the depth of the hole below the water surface be $x$. The height of the hole above the ground is $(h - x + H)$.The velocity of efflux is $v = \sqrt{2gx}$.The time taken for the water to reach the ground is $t = \sqrt{\frac{2(h - x + H)}{g}}$.The horizontal range $R$ is given by $R = v \cdot t = \sqrt{2gx} \cdot \sqrt{\frac{2(h - x + H)}{g}} = 2\sqrt{x(h + H - x)}$.For maximum range, the expression inside the square root, $f(x) = x(h + H - x) = x(h + H) - x^2$, must be maximized.Taking the derivative with respect to $x$ and setting it to zero: $\frac{df}{dx} = (h + H) - 2x = 0$.Thus, $x = \frac{h + H}{2}$.

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