Let us now imagine population flux through Isla Nublar $($the island where Jurassic Park is located$).$ Visitors

are transported in and out on cable cars. We can model this as a mixing problem! Suppose the island has

$100$ cable cars and $10000$ visitors at the beginning of the day, with $5$ cable cars arriving and $10$ cable cars

departing every hour. Assume that each arriving cable car contains $300$ people, and each departing flight

carries the average number of people per available cable car on the island at that moment. $($We will assume

that $t$ is measured in hours.$)$ Finally, some of the visitors on the island decide to take a ferry to the nearby

Muertes Archipelago. We will assume that the rate of such departures is $10(t+\frac{1}{1+t})$ visitors per hour.

$($a$)$ Setup the initial value problem for the number of visitors on the island, $x(t).$ Hint: To draw an analogy

to the lectures, the cable cars are the water and the people are the salt. Don't forget that this "tank" is

leaking some "salt$-$water solution" to the archipelago nearby!

$($b$)$ Next, solve the mixing problem using the "ode$45"$ function on MATLAB, over the time interval $0,15,$

i$.$e$,$ for the first $15$ hours of operation. Plot the solution $x(t)$ against $t,$ and make sure you label the axes.

$($c$)$ Finally, using a "for" loop, and Euler's method described below, find the solution to the mixing problem

$($d$)$ Repeat part $($c$)$ with time$-$steps of $15,30,60$ minutes. $($Keep in mind that $t$ is measured in hours!$)$ Plot

all these solutions on the same figure as the one containing part $($c$).$ Use different colors for the curves,

and include a "Legend" to identify them. $($If the curves are indistinguishably close to each other, use

some other values for the time$-$steps.$)$

$($e$)$ Compare the plots from parts $($c$)$ and $($d$)$ to the one from $($b$),$ which is considered to be the "exact"

solution. What happens to the solutions in $($c$)$ and $($d$)$ as you change the time$-$step?