An oil spill has fouled $200$ miles of Pacific shoreline. The oil company responsible has been given $14$ days to clean up, after which a fine will be $10000$ \$/day. The local cleanup crew cleans $5$ miles/day at a cost of $500$ \$/day. Additional crews can be $18000$ dollars plus $800$ \$/day.
How many additional crews should be brought to minimize the total cost? How much is cost?
So far, I have the variables
- $t$ - time in days to clean up the oil spill
- $n$ - total crew
- $n_0$ - number of crew to clean up in exactly 14 days
- $p$ - the amount of fine in dollars
- $C$ - total cost in dollars
- $C$ =
I am not sure how to model the assumptions to minimize the cost or how to model the fine. Please help!
$$\begin{align} 200 &= 5t + 5tn \\ C &= \begin{cases} 18000n + 800tn + 500t \quad\quad\quad\quad\quad\quad\quad\quad\mathrm{if}\quad t \leq 14 \\ 18000n + 800tn + 500t + 1000(t-14) \quad\quad\mathrm{if}\quad t > 14 \end{cases} \end{align}$$
Solve the first equation for $n$. Substitute it into the second equation to get a piecewise function only in terms of $t$. Take the derivative of each piece $C(t)$ and find the zeros. If any those zeros are not integers, consider instead the integers on either side of that zero (we need a whole number of days). Plug these values (the zeros or their nearest integers) back into $C(t)$ to find the minimum. Use that $t$ in the first equation and solve for $n$.