Question

using production and geological data, the management of an oil company estimates that oil will be pumped from a producing field at a rate given by the following.

r(t)=\frac{80}{t + 8}+8;0\leq t\leq15

r(t) is the rate of production (in thousands of barrels per year) t years after pumping begins. find the area between the graph of r and the t - axis over the interval 6,12 and interpret the results.

the area is approximately 77 square units. (round to the nearest integer as needed)

choose the correct interpretation of the results below.

○ a. it will take approximately 77 years after pumping begins to reach a thousand barrels.

○ b. it will take approximately 78 years after pumping begins to reach a thousand barrels.

○ c. total production from the end of the sixth year to the end of the twelfth year will be approximately 77 thousand barrels.

○ d. total production from the end of the first year to the end of the fifteenth year will be approximately 77 thousand barrels.

Explanation:

The area between the graph of a rate - function \(R(t)\) (where \(R(t)\) is the rate of production in thousands of barrels per year) and the \(t\) - axis over an interval \([a,b]\) represents the total production over that time interval. Here, the interval is \([6,12]\).

Since \(R(t)\) is the rate of production (in thousands of barrels per year) \(t\) years after pumping begins, integrating \(R(t)\) from \(t = 6\) to \(t=12\) (finding the area between the graph of \(R\) and the \(t\) - axis over \([6,12]\)) gives the total production over the time period from \(t = 6\) to \(t = 12\).

Answer:

C. Total production from the end of the sixth year to the end of the twelfth year will be approximately 77 thousand barrels.