Final answer:
To find the number of additional wells needed for maximum oil production, create a quadratic function from the given information. Calculate the vertex of the parabola, which indicates the optimal number of additional wells, resulting in 6 additional wells needed to maximize daily oil output.
Step-by-step explanation:
To determine how many additional wells should be drilled to obtain the maximum amount of oil per day, we can formulate an equation based on the given information. Initially, there are 8 wells producing a total of 1600 barrels per day, which gives an average of 200 barrels per well per day. If for each additional well the average production decreases by 10 barrels, then for x additional wells, the average production per well will be 200 - 10x. The total production P(x) with the additional wells can be represented by:
P(x) = (8 + x)(200 - 10x)
To maximize the total production P(x), we need to find the vertex of this quadratic function, since the leading coefficient is negative, which ensures the parabola opens downward and thus the vertex will give the maximum value. The vertex form of a quadratic function is P(x) = a(x - h)² + k, where (h, k) is the vertex of the parabola. We can find the vertex by completing the square or by using the vertex formula h = -b/(2a) for a quadratic equation in the form of ax²+ bx + c.
By expanding the equation for total production and comparing it to the standard quadratic form, we get:
P(x) = -10x² + (200 - 80)x + 1600
The value of h gives the number of additional wells for maximum production. Therefore:
h = -b/(2a) = -(200 - 80)/(2 * -10) = 6
Since we cannot have a fraction of a well, the maximum amount of oil per day would be obtained by drilling 6 additional wells.