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?
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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.
actually, you want only 6 more than the existing 8 wells.
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. ☺☺☺☺
Well, well, well, looks like we have an oil conundrum here. Let's drill down and find the solution!
Your equation looks correct, but it seems there might be a mix-up with the derivative. To find the maximum, we need to find the number of wells that maximizes the total production, which is the product of the number of wells and the average production per well.
Let's review what you started with: x(1600 - (x-8)10), where x is the number of wells. Now, let's simplify it a bit: x(1600 - 10(x-8)).
To find the maximum, we need to take the derivative of this equation with respect to x and set it equal to zero. Let's do some math!
d/dx[x(1600 - 10(x-8))] = 0
1600 - 10(x-8) - 10x = 0
1600 - 10x + 80 - 10x = 0
1600 - 20x + 80 = 0
1680 - 20x = 0
20x = 1680
x = 1680/20
x = 84
So it looks like your answer of 84 additional wells is correct after all! Sometimes the answer key can be a bit of a joker. Keep drilling, my friend, and enjoy the oil boom!
To find the optimal number of additional wells to maximize the amount of oil produced per day, we need to maximize the total production function. Let's start by redefining the given information:
Let x be the number of additional wells to be drilled.
Each additional well decreases the average production per well by 10 barrels per day.
The total production per day can be written as:
Total production = (8 + x) * (1600 - 10x)
To maximize the total production, you can take the derivative with respect to x and set it equal to 0, then solve for x:
d(Total production)/dx = (1600 - 10x) + (8 + x) * (-10) = 0
Simplifying the equation:
1600 - 10x - 10x - 80 - 10x = 0
1600 - 30x - 80 = 0
1520 - 30x = 0
-30x = -1520
x = 1520 / 30
x ≈ 50.67
Since x represents the number of additional wells, it cannot be a fractional value. Hence, rounding up x gives us the optimal number of additional wells to be drilled to obtain the maximum amount of oil per day:
x ≈ 51
Therefore, you should drill an additional 51 wells to obtain the maximum amount of oil per day.
To solve this problem, you correctly set up the equation as x(1600 - (x-8) * 10), where x represents the total number of wells (including the original 8 wells).
However, the derivative and setting it equal to zero is not the correct approach in this case. Since we want to find the maximum amount of oil per day, we need to find the value of x that gives us the maximum value of the equation.
To determine the maximum value, we can graph the equation and observe the behavior of the function.
Let's plot the graph:
First, we simplify the equation: x(1600 - 10x + 80) = x(1680 - 10x).
To graph, plot the equation on a graphing tool or software, or manually calculate the points for different values of x. Here are a few points:
x = 8: 8(1680 - 10*8) = 8 * 1600 = 12800
x = 9: 9(1680 - 10*9) = 9 * 1590 = 14310
x = 10: 10(1680 - 10*10) = 10 * 1480 = 14800
x = 11: 11(1680 - 10*11) = 11 * 1370 = 15070
x = 12: 12(1680 - 10*12) = 12 * 1260 = 15120
By observing the behavior of the function, we can see that it initially increases as we add more wells (up to a certain point) and then starts to decrease. We are interested in finding the point where the function reaches its maximum value.
From the points plotted, we can see that the maximum value occurs at x = 12, where the production is 15120 barrels of oil per day.
Therefore, to obtain the maximum amount of oil per day, you should drill an additional 12 - 8 = 4 wells.
So, the correct answer is 4 additional wells, not 84 as you previously calculated.
Remember, graphing the function helps us visually understand the behavior and identify the maximum value in this case.