Part $1$: Draining a Cylindrical Tank

A$)$ Suppose a cylindrical tank with a drainage valve at its base is full of water so that the

initial water level is the height of the tank, which we will write as $y_0($in meters$).$ The

tank has a diameter of $d_{tank}($in meters$)$ and its drainage valve has a diameter of $d_{\text{v}\text{a}\text{l}\text{v}\text{e}}($in

meters$).$ When the drainage valve opens, water exits the tank, causing the water level to

drop until the tank is empty. Let the water level in the tank be $y=y(t),($in meters$),$ where

$t$ represents time $($in seconds$).$ The mathematical model for this system is given below as

a first order differential equation.

$\{\begin{array}{c}\text{:}[A=\sqrt[\phantom{2}]{13}]\\ B=0\mathrm{.}35\end{array},\frac{dy}{dt}=-\frac{(d_{\text{v}\text{a}\text{l}\text{v}\text{e}}{)}^2}{(d_{tank}{)}^2}\sqrt[\phantom{2}]{2gy},y(0)=y_0$

Solve this IVP by hand using separation of variables. Show all steps and provide all

relevant work. From the solution of this differential equation, determine how long it will

take for the tank to drain. Use $g=9\mathrm{.}8\frac{m}{s^2},d_{tank}=A,d_{\text{v}\text{a}\text{l}\text{v}\text{e}}=B,$ and $y_0=2A.$

B$)$ In MATLAB, write a script that creates a direction field for this differential equation.

Describe why the direction field looks the way it does. Provide the plot and the code

used in your final submission.

C$)$ In MATLAB, write a script that applies Euler's, Improved Euler's, and the Runge$-$

Kutta $($RK$4)$ numerical methods for the drainage of the cylindrical tank. Plot all three

results onto a single set of axes. Use different colors for each plot to distinguish between

them and use a step size no larger than $t=0\mathrm{.}5.$ Provide the plot and the code used in

your final submission.