An oil field contains 8 wells, which produce a total of 1600 barrels of oil per day. For each additional well that is drilled, the average production per well decreases by 10 barrels per day. How many additional wells should be drilled to obtain the maximum amount of oil per day?

Question

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I set up the equation x(1600-(x-8)10), where x is the number of wells, took the derivative, and set the derivative equal to zero to find the optimal number of additional wells to build. I got 84, but the answer key said I was wrong.

Answer 1

Total production before = 8*200 barrels
Ok, so we want to find the maximum possible production. To do this, we have to find a function for the amount of barrels produced by x wells. We know that the original average for 8 wells is 200 barrels per well. The equation relating the total production to number of wells is T(x)=average*x. Since we know the average is given by 200-(10*(x-8)), we can write this equation as T(x)=x(200-10(x-8))=x(200-10x+80)=-10x²+280x. Now, in order to find a maximum, we take the derivative and set it equal to zero. So:
-20x+280=0
20x=280
x=14
In order to maximize production, you need to drill 14 additional wells. ☺☺☺☺

Answer 2

actually, you want only 6 more than the existing 8 wells.