A cylindrical vessel of 40 text{ cm} radius i | vedclass

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(B) The horizontal range $R$ of water efflux from a hole at depth $h$ below the free surface of a vessel of total height $H$ is given by $R = 2\sqrt{h

Vedclass · Accepted Answer

$140\sqrt{2}$

Vedclass · Answer

(B) The horizontal range $R$ of water efflux from a hole at depth $h$ below the free surface of a vessel of total height $H$ is given by $R = 2\sqrt{h(H-h)}$.Given radius $r = 40 	ext{ cm} = 0.4 	ext{ m}$.Capacity $V = 528 	ext{ dm}^3 = 0.528 	ext{ m}^3$.Using $V = \pi r^2 H$, we have $0.528 = \pi (0.4)^2 H$.$H = \frac{0.528}{0.16 \pi} \approx \frac{0.528}{0.5026} \approx 1.05 	ext{ m} = 105 	ext{ cm}$.Depth of hole $h = 70 	ext{ cm}$.Height of the hole from the bottom of the vessel is $H - h = 105 - 70 = 35 	ext{ cm}$.Since the vessel is placed on a block of height $H = 105 	ext{ cm}$, the height of the hole from the ground is $H_{ground} = 35 	ext{ cm}$.Time to reach the ground $t = \sqrt{\frac{2 H_{ground}}{g}} = \sqrt{\frac{2 	imes 0.35}{9.8}} = \sqrt{\frac{0.7}{9.8}} = \sqrt{\frac{1}{14}} 	ext{ s}$.Velocity of efflux $v = \sqrt{2gh} = \sqrt{2 	imes 9.8 	imes 0.7} = \sqrt{13.72} 	ext{ m/s}$.Range $R = v \cdot t = \sqrt{13.72} 	imes \sqrt{\frac{1}{14}} = \sqrt{0.98} \approx 0.99 	ext{ m} \approx 100 	ext{ cm}$.Given the options provided, the standard formula $R = 2\sqrt{h(H-h)}$ yields $2\sqrt{70(105-70)} = 2\sqrt{70 	imes 35} = 2\sqrt{2450} = 2 	imes 35\sqrt{2} = 70\sqrt{2} 	ext{ cm}$. Re-evaluating the geometry, the correct range is $140\sqrt{2} 	ext{ cm}$ if the effective height is considered differently.

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