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(D) Let $H$ be the total height of the water column. Let $h^{\prime}$ be the depth of hole $B$ from the top surface and $h$ be the height of hole $A$

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$1$

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(D) Let $H$ be the total height of the water column. Let $h^{\prime}$ be the depth of hole $B$ from the top surface and $h$ be the height of hole $A$ from the bottom.For hole $B$ at depth $h^{\prime}$ from the top,the velocity of efflux is $v_B = \sqrt{2gh^{\prime}}$. The height of this hole from the ground is $H - h^{\prime}$. The time taken for water to reach the ground is $t_B = \sqrt{\frac{2(H - h^{\prime})}{g}}$.The range $R_B = v_B 	imes t_B = \sqrt{2gh^{\prime}} 	imes \sqrt{\frac{2(H - h^{\prime})}{g}} = 2\sqrt{h^{\prime}(H - h^{\prime})}$.For hole $A$ at height $h$ from the bottom,the depth from the top is $H - h$. The velocity of efflux is $v_A = \sqrt{2g(H - h)}$. The time taken to reach the ground is $t_A = \sqrt{\frac{2h}{g}}$.The range $R_A = v_A 	imes t_A = \sqrt{2g(H - h)} 	imes \sqrt{\frac{2h}{g}} = 2\sqrt{h(H - h)}$.For the same range,$R_A = R_B$:$2\sqrt{h(H - h)} = 2\sqrt{h^{\prime}(H - h^{\prime})}$$h(H - h) = h^{\prime}(H - h^{\prime})$$hH - h^2 = h^{\prime}H - h^{\prime 2}$$H(h - h^{\prime}) = h^2 - h^{\prime 2}$$H(h - h^{\prime}) = (h - h^{\prime})(h + h^{\prime})$Since $h 
eq h^{\prime}$,we have $H = h + h^{\prime}$.This implies that the two holes are at depths $h^{\prime}$ and $H - h^{\prime} = h$ from the top surface. For the same range,the sum of the depths of the two holes from the top surface must be equal to the total height of the water column. In the given configuration,if $h^{\prime}$ is the depth of $B$ and $h$ is the height of $A$ from the bottom,the condition for equal range is $h = h^{\prime}$,so the ratio $\frac{h^{\prime}}{h} = 1$.

$A$ tank is filled with water and two holes $A$ and $B$ are made in it. For getting the same range,the ratio of $h^{\prime}$ to $h$ is ..........

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