Problem $1$

Consider a lake with volume of $100000m^3.$ Each day $1000m^3$ of water flows into the lake,

which is partially clean water, partially pollution. On day $0$ there is no pollution in the

lake. Suppose that inflow consists of $5m^3$ of pollution and $995m^3$ of clean water daily. The

outflow from the lake is $1000m^3$ daily.

$($a$)$ Construct a difference equation describing this process $($see hint on the next page $-$

the equation is construted there, you only need to substitute numerical data$).$

$($b$)$ Solve this difference equation, and obtain formula for the level of pollution as a

function of time.

$($c$)$ Write Python program plotting the level of pollution in the lake as a function of time.

The easiest way to do this is to modify program illustrating the growth of city's population,

supplied on the SAKAI page.

Verify correctness of your program by checking that your solution obtained in $($b$)$ agrees

with results obtained from the program. The easies way to do it is to calculate, using

formula from $($b$),$ the pollution level after, for example, $30$ days, and then check what the

program outputs after $30$ days. You do not have to submit anything for part $($c$)-$ this is

just verification of correctness of your program.