An oil spill has fouled 250 miles of Pacific shoreline. The company responsible has been ordered to clean the mess. Each clean-up crew costs $9000 to bring in, and $300 per crew per day. Each crew can clean 1 mile of shoreline per day. To encourage a speedy cleanup, the government is levying a fine of $1000 per day against the company.Your group has been hired by the company to determine how many crews the company should hire to minimize their costs.
1. Find an equation that gives the cost, C of cleanup in terms of the number of crews, x hired: C(x)=
2. Find the derivative of the cost equation C'(x)=
3. Determine the number of crews that should be brought in to minimize the cost to the company.
a. Mathematically, the number of crews would be:
Give the mathematical answer to at least two decimal places.
b. Since partial crews cannot be brought in, the whole number of crews that should be brought in to minimize cost the company would be:
c. The minimum cost to the company (using your answer from 3b) will be: $
Note: The answer to 3b may not be as simple as rounding the answer to 3a. You should check the actual cost to the company to the whole numbers above and below your answer to 3a.
5. Rewrite your equation from above, replacing the $1000 fine with the constant f. Your equation will involve both the variable x and the constant f: C(x)=
6. Determine the number of crews that should be brought into minimize the cost to the company as a function of the fine amount.