A) \[\sqrt{3}\]
B) \[\sqrt{2}\]
C) \[\frac{2-\sqrt{2}}{\sqrt{2}}\]
D) \[\frac{2-\sqrt{3}}{\sqrt{3}}\]
Correct Answer: D
Solution :
[d] Let A and a be cross sectional areas of the tank and hole respectively. Let h be height of water in the tank at a time t. \[Let\,\left[ -\frac{dh}{dt} \right]\] represent the rate of fall of level. \[-\frac{Adh}{dt}=a\sqrt{2gh}\] \[dt=\frac{-A}{a\sqrt{2g}}\frac{dh}{\sqrt{h}}\] Ratio \[\frac{{{t}_{1}}}{{{t}_{2}}}=\frac{\int\limits_{0}^{3{}^{h}/{}_{4}}{\frac{dh}{\sqrt{h}}}}{\int\limits_{3{}^{h}/{}_{4}}^{0}{\frac{dh}{\sqrt{h}}}}=\frac{\sqrt{\frac{3h}{4}}-\sqrt{h}}{0-\sqrt{\frac{3h}{4}}}=\frac{2-\sqrt{3}}{\sqrt{3}}\]
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