# word Problem

• Jan 21st 2009, 01:13 AM
justinwager
word Problem
1) An oil field containings 20 wells has been producing 4000 barrels of oil daily. For each new well that is drilled, the daily production of each well decreases by 5 barrels.

i) Write the total daily production of the oil field as a function of the number x of new wells drilled.

ii) What number x of new wells drilled will maximize the total daily production of oil.

Now I don't have any idea at all how to do this problem...but i realize it will be a parabla function as it asks for a maxmium....Also how could i find the maximum without using a graphing calculator when the equastion?

thanks for the help guys!
• Jan 21st 2009, 01:41 AM
Gusbob
Quote:

Originally Posted by justinwager
1) An oil field containings 20 wells has been producing 4000 barrels of oil daily. For each new well that is drilled, the daily production of each well decreases by 5 barrels.

i) Write the total daily production of the oil field as a function of the number x of new wells drilled.

ii) What number x of new wells drilled will maximize the total daily production of oil.

The first thing we need to do is to make this into an equation. So assuming each well has the same amount of production, 4000 barrels / 20 wells = 200 barrels per well. Our current formula looks like this:
Production = Wells x Barrels per well (4000 = 20 x 200)

Lets say production = y and new wells = x
Since for every new well, we get a decrease of 5 barells
Production = (Wells + New wells) x (Barells per well - 5 barells per new well)

Mathematically, it looks like this:

$\displaystyle y = (20 + x) (200 - 5x)$
$\displaystyle = -5x^2 + 100x + 4000$

Differentiate this function and equate it to 0 to find the maximum point, keeping in mind that x indicates # of new wells, not all wells.
• Jan 21st 2009, 02:11 PM
justinwager
thanks i figured out how you did that part but im unclear what you by equate by 0 to find the maximum? :S

Thanks!
• Jan 21st 2009, 05:21 PM
justinwager
bump
• Jan 21st 2009, 09:41 PM
justinwager

thanks!
• Jan 22nd 2009, 12:03 AM
Gusbob
Since you posted this under calculus, I assume you have some knowledge about finding maximum points of functions using differentiation?

If not:
The function we derived is obviously a parabola with a maximum point (maximum y, in this case production).

In any parabola $\displaystyle ax^2 + bx + c$, the x value for the sole maximum/minimum point is given by
$\displaystyle x = \frac{-b}{2a}$

Use this formula on the derived function (where b = 100, a = -5) to find x, the number of new wells required to maximise production
• Jan 22nd 2009, 12:43 AM
justinwager
omg....I TOTALLY forgot this formula...thanks so much mate!

Cheers!