$($PLEASE DO NOT COPY ANYONE ELSES WORK!!! AND SOLVE ALL PARTS OF THE PROBLEM!!!$)$

A small reservoir has a constant area $A,$ and the water in the reservoir has a constant density

$.$ The reservoir has a small opening where the water can flow out. When the water flows out

of the reservoir, some momentum is lost. Hence, we can characterize the loss by a constant

resistance coefficient $R.$ In addition to the outflow, the reservoir also receives various water

inflows, such as rain. Let us assume that the flow rate $($fluid volume per unit time, e$.$g$.,$

$\{$:$\frac{m^3}{s})$ into the reservoir is $Q(t).$ Then the water level $h$ of the reservoir satisfies the following

first$-$order differential equation

$A\frac{dh(t)}{dt}+\frac{g}{R}h(t)=Q(t)$

where $g$ is the gravitational acceleration. Based on $(1),$ answer the following questions.

$($a$)$ What is the time constant of the reservoir?

$($b$)$ How much inflow $Q(t)$ is needed to maintain a steady $($i$.$e$.,$ constant$)$ water level $\frac{?}{\text{b}\text{a}\text{r}}(h)$ of

the reservoir?

$($c$)$ A big storm came one day, and $m^3$ of water is flooded into the reservoir in a very short

period of time. How do you approximate $Q(t)$ mathematically? In other words, what

would be a good way to describe $Q(t)$ mathematically?

$($d$)$ If the water level before the storm is $h_0,$ what is the water level immediate after the

storm?

$($e$)$ After the storm is over, predict how the water level changes with respect to time.PLWA