I was given a question involving creating a function for the height of the water level in a tank, given gravity, the size of the hole in the tank and dimensions of the tank, as identified here.
I am completely unsure as to where to start for this problem, and possibly the correct answer will help me backtrack as to how to approach the question.
You need to relate the volume of water to depth and time. To relate to depth, if we look at a circular cross section of the tank with the origin at the bottom and the $y$-axis pointing up, the equation for the circle is $x^2+(y-R)^2=R^2$ where $R$ is the tank's radius. Then the for a given depth $y$, $x=\pm\sqrt{R^2-(y-R)^2}=\pm\sqrt{2Ry-y^2}$, so the width of the surface of the water in the tank is $2\sqrt{2Ry-y^2}$ and its length is $L$, so if we piled water on top with extra thickness $dy$ the extra volume would be $dV=L\sqrt{2Ry-y^2}dy$. We have the relationship between volume and depth that we wanted, $$\frac{dV}{dy}=L\sqrt{2Ry-y^2}$$ Assuming water flows without viscous losses it should follow Bernoulli's equation $$p_1+\frac12\rho v_1^2+\rho gy_1=p_2+\frac12\rho v_2^2+\rho gy_2$$ Where $p_1=p_2=P_{\text{atmospheric}}$ for both the surface of the water and for the free-flowing stream of water. Also the water is moving very slowly at the surface in the tank, so take $v_1=0$. Then $y_1-y_2=y$ is the depth of water in the tank, so we are down to $$\rho gy=\frac12\rho v^2$$ with $v_2=v$, the speed of the stream flowing out of the hole. In a small period of time $dt$ a volume of water with cross-sectional area $B$ and length $v\,dt$ flows out of the hole so $$dV^{\prime}=Bv\,dt=B\sqrt{2gy}dt$$ flows out of the hole so the rate of increase of volume is $$\frac{dV}{dt}=-\frac{dV^{\prime}}{dt}=-B\sqrt{2gy}$$ Knowing $\frac{dV}{dy}$ and $\frac{dV}{dt}$ we can find $\frac{dy}{dt}$ and integrate. The solution comes out as $$t_e=\frac{2L}{B\sqrt{2g}}\left(\frac{12\,\text{in}}{1\,\text{ft}}\right)^2\cdot\frac23R^{3/2}\left(2^{3/2}-1\right)\approx18502.4\,\text{sec}$$ By my calculation.