Chapter 1, Problem 9P

A storage tank contains a liquid at depth y, where $y=0$ when the tank is half full. Liquid is withdrawn at a constant low rate Q to meet demands. The contents are resupplied at a sinusoidal rate $3Q{\text{sin}}^{\text{2}}\left(t\right)$.

FIGURE P1.9

Equation (1.13) can be written for this system as

$\begin{array}{l}\frac{d\left(Ay\right)}{dt}=3Q{\sin }^2\left(t\right)-Q\\ \left(\text{change in volume}\right)=\left(\text{inflow}\right)-\left(\text{outflow}\right)\end{array}$

or, since the surface area A is constant

$\frac{dy}{dt}=3\frac{Q}{A}{\sin }^2\left(t\right)-\frac{Q}{A}$

Use Euler's method to solve for the depth y from $t=0$ to 10 d with a step size of 0.5 d. The parameter values are $A=1250{\text{ m}}^{\text{2}}\text{ and }Q=450{\text{ m}}^{\text{3}}\text{/d}$. Assume that the initial condition is $y=0$.