Problem Description $(CCOs$#$1,2,3,4,5,6,7,8,11,12)$

A water tank of radius $R=1\mathrm{.}8m$ with two outlet pipes of radius $r_1=0\mathrm{.}05m$ and $r_2$ installed at heights $h_1=0\mathrm{.}13m$ and $h_2=1m,$ is mounted in an elevator moving up and down causing a time dependent acceleration $g(t)$ that must be modeled as

$g(t)=g_0+a_{1}cos(2f_{1}t)+b_{1}sin(2f_{1}t)+a_{2}cos(2f_{2}t)+b_{2}sin(2f_{2}t),$

Figure $1$: Water tank inside an elevator

The height of water $h(t)$ in the tank can be modeled by the following ODE,

$\frac{dh}{dt}=\frac{f(t)-\sqrt[\phantom{2}]{2g(t)}(r_1^2\sqrt[\phantom{2}]{\text{m}\text{a}\text{x}(0,h-h_1)}+r_2^2\sqrt[\phantom{2}]{\text{m}\text{a}\text{x}(0,h-h_2)})}{R^2}$

where $=1000k\frac{g}{m^3}.$ The volume flow rate $V^{}(t)$ of water out of the tank is

$V^{}(t)=\sqrt[\phantom{2}]{2g(t)}(r_1^2\sqrt[\phantom{2}]{\text{m}\text{a}\text{x}(0,h(t)-h_1)}+r_2^2\sqrt[\phantom{2}]{\text{m}\text{a}\text{x}(0,h(t)-h_2)})$

To help determine the model constants in Eq$.(1),$ measurements of the elevator position $y(t)$ are taken every $5$ s starting at $t=0s$ until $2000$s$.$ The measured position data is available on Canvas in the Matlab file ydat $.$ mat. The mass flow rate $f(t)$ in $k\frac{g}{s}$ into the tank is measured every $10$ s starting at $0$ s and ending at $2500$ s and is available on Canvas in the Matlab file fdat. mat. At $t=0s,$ the height of water in the tank is