A spherical tank with negligible thickness and internal radius $a$cm contains water. At time $t$

Question

A spherical tank with negligible thickness and internal radius $a$cm contains water. At time $t$s, the water surface is at a height $x$cm above the lowest point of the tank and the volume of water in the tank, $V$cm$^{3}$, is given by $V=\frac{1}{3}\pi {{x}^{2}}\left( 3a-x \right)$. Water flows from the tank, through an outlet at its lowest point, at a rate $\pi k\sqrt{x}$cm$^{3}$ s$^{-1}$, where $k$ is a positive constant. Show that $\left( 2ax-{{x}^{2}} \right)\frac{\mathrm{d}x}{\mathrm{d}t}=-k\sqrt{x}$.[2] Find the general solution for $t$ in terms of $x$,$a$ and $k$.[3] Find the ratio ${{T}_{1}}:{{T}_{2}}$, where ${{T}_{1}}$ is the time taken to empty the tank when initially it is completely full, and ${{T}_{2}}$ is the time taken to empty the tank when initially it is half full.[4]

Accepted Answer

(ii) $t=\frac{1}{k}\left[ c-\frac{4}{3}a{{x}^{\frac{3}{2}}}+\frac{2}{5}{{x}^{\frac{5}{2}}} \right]$ (iii) Ratio$=8\sqrt{2}:7$

2019 NYJC P1 Q7

2019 NYJC P1 Q7

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