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(D) The range of efflux $x$ for a hole at height $h$ from the bottom in a vessel filled with liquid to a total height $H$ is given by $x = 2 \sqrt{(H

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$\frac{h_2 + h_1}{2}$

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(D) The range of efflux $x$ for a hole at height $h$ from the bottom in a vessel filled with liquid to a total height $H$ is given by $x = 2 \sqrt{(H - h)h}$.Given that the range of efflux is the same for two holes at heights $h_1$ and $h_2$,we have:$2 \sqrt{(H - h_1)h_1} = 2 \sqrt{(H - h_2)h_2}$$(H - h_1)h_1 = (H - h_2)h_2$$Hh_1 - h_1^2 = Hh_2 - h_2^2$$H(h_1 - h_2) = h_1^2 - h_2^2$$H(h_1 - h_2) = (h_1 - h_2)(h_1 + h_2)$$H = h_1 + h_2$The range of efflux $x = 2 \sqrt{Hh - h^2}$ is maximum when the term inside the square root is maximum. Let $f(h) = Hh - h^2$. For maximum,$f'(h) = 0$.$H - 2h = 0 \Rightarrow h = \frac{H}{2}$.Substituting $H = h_1 + h_2$,we get the height for maximum range as $h = \frac{h_1 + h_2}{2}$.

In a cylindrical vessel containing liquid of density $\rho$,there are two holes in the side walls at heights of $h_1$ and $h_2$ respectively,such that the range of efflux at the bottom of the vessel is the same. The height of a hole,for which the range of efflux would be maximum,is:

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