Based on a preliminary report by a geological survey team, it is estimated that a newly discovered oil field can be expected to produce oil at the rate of
R(t) =
300t^2/(t^3 + 32)+5 (0 ≤ t ≤ 20)
thousand barrels/year, t years after production begins. Find the amount of oil that the field can be expected to yield during the first 3 years of production, assuming that the projection holds true. (Round your answer to the nearest thousand barrels.)
that would be
∫[0,3] 300t^2/(t^3+32) + 5 dt
let
u = t^3 + 32
du = 3t^2 dt
and now you have
∫[32,59] 100/u du + ∫[0,3] 5 dt
= 15 + 100log(59/32)
76
To find the amount of oil that the field can be expected to yield during the first 3 years of production, we need to calculate the integral of the production rate function, R(t), over the interval from 0 to 3.
The integral of R(t) can be calculated as follows:
∫[0,3] R(t) dt = ∫[0,3] (300t^2/(t^3 + 32) + 5) dt
Integrating the first term requires some algebraic manipulation.
Let's start by simplifying the denominator of the first term:
t^3 + 32 = (t + 2)(t^2 - 2t + 4)
Now we can rewrite the first term of the integrand:
300t^2/(t^3 + 32) = 300t^2/[(t + 2)(t^2 - 2t + 4)]
We can proceed to partial fraction decomposition to simplify further. The denominator can be factored as:
(t + 2)(t^2 - 2t + 4) = (t + 2)(t - 1 + i√3)(t - 1 - i√3)
Now we can write the partial fraction decomposition:
300t^2/[(t + 2)(t^2 - 2t + 4)] = A/(t + 2) + (Bt + C)/(t^2 - 2t + 4) + (Dt + E)/(t^2 - 2t + 4)
Solving for the constants A, B, C, D, and E gives us the following values:
A = 20
B = 160
C = 120
D = 16
E = -56
Now we can rewrite the integral as:
∫[0,3] (20/(t + 2) + (160t + 120)/(t^2 - 2t + 4) + (16t - 56)/(t^2 - 2t + 4)) dt
Evaluating the integral from 0 to 3 using these expressions will give us the amount of oil that the field can be expected to yield during the first 3 years of production.
∫[0,3] (20/(t + 2) + (160t + 120)/(t^2 - 2t + 4) + (16t - 56)/(t^2 - 2t + 4)) dt = [950.8]
Therefore, the field can be expected to yield approximately 951,000 barrels of oil during the first 3 years of production, assuming the projection holds true.
To find the amount of oil the field can be expected to yield during the first 3 years of production, we need to calculate the integral of the production rate function over the interval [0, 3]. The integral will give us the total oil produced during this time period.
First, let's find the indefinite integral of the production rate function R(t).
∫ [300t^2/(t^3 + 32) + 5] dt
To solve this integral, we use the substitution method. Let u = t^3 + 32. Then, du = 3t^2 dt. Rearranging this equation, we have dt = du/(3t^2).
∫ [300t^2/(t^3 + 32) + 5] dt
= ∫ [300t^2/u + 5] (du/(3t^2))
= ∫ [100/u + 5/3] du
Now we can integrate each term separately:
∫ (100/u + 5/3) du
= 100 * ln|u|/3 + (5/3)u + C
Replacing u with t^3 + 32 and simplifying the expression:
= 100 * ln|t^3 + 32|/3 + (5/3)(t^3 + 32) + C
Next, we evaluate the definite integral over the interval [0, 3]:
∫[0,3] (100 * ln|t^3 + 32|/3 + (5/3)(t^3 + 32)) dt
Substituting the upper and lower limits into the expression:
= [100 * ln|3^3 + 32|/3 + (5/3)(3^3 + 32)] - [100 * ln|0^3 + 32|/3 + (5/3)(0^3 + 32)]
= [100 * ln(35)/3 + (5/3)(35)] - [100 * ln(32)/3 + (5/3)(32)]
Now we can calculate this expression to find the amount of oil expected to be produced during the first 3 years of production.