Optimizing Oil Spill Cleanup: Minimizing Costs with Additional Crews

[SMA_01]

An oil spill has fouled 200 miles of shoreline. The oil company responsible has been given 14 days to clean up the shoreline, after which a fine of $$10/day$ will be imposed. The local clean up crew can scrub 5 miles of beach per week at a cost of $500/day. Additional crews can be brought in at a cost of $18 plus $800/day for each crew. How many additional crews should be brought into minimize the total cost to the company?

I'm supposed to use these variables:

n= total number of crews, including the local crew

n_0=number of crews required to clean up the shoreline in exactly 14 days

t= number of days to clean up the oil spill

c= total cost (measured in thousands of dollars)

t=the amount of the fine (measure in thousands of dollars)

I'm stuck on this, I am having difficulty putting the pieces together. I'm supposed to find a formula for t in terms of n, t(n).

How can I relate the number of days to finish the job to the number of crews? How can I find the number of additional crews with my given variables?

I know that if t>14 or n<n0, then they will have to pay a fine...

 

[tiny-tim]

Hi SMA_01!

Do it step-by-step …

call the number of crews "n"

write the equation for the time it will take (as a function of n)

then write the equation for the cost (as a function of n) …

show us what you get. ​

 

[SMA_01]

tiny-tim,

Thanks, I got

t(n)=280/n

c(t)=500t+(18+800t)(n-1), n>=20
c(t)=500t+(18+800t)(n-1)+10(t-14)

 

[tiny-tim]

Hi SMA_01!

That looks good.

Now substitute your formula for t(n) into your two formulas for c(n).

(and don't forget the 280/n has to rounded up to the nearest whole number)