Problem 49 - Applied Calculus

Question

BUSINESS: Oil Well Output An oil well is expected to produce oil at the rate of \(50 e^{-0.05 t}\) thousand barrels per month indefinitely, where \(t\) is the number of months that the well has been in operation. Find the total output over the lifetime of the well by integrating this rate from 0 to \(\infty\). [Note: The owner will shut down the well when production falls too low, but it is convenient to estimate the total output as if production continued forever.]

Step-by-Step Solution

Answer

The total oil output is 1000 thousand barrels.

Step 1: Understand the Problem

We need to find the total amount of oil produced by the well from month 0 to infinity, which involves integrating the rate of production \(50 e^{-0.05 t}\) from 0 to \(\infty\).

Step 2: Set Up the Integral

The formula for the total output is the integral \( \int_0^{\infty} 50 e^{-0.05 t} \, dt \). We need to evaluate this integral.

Step 3: Simplify the Integral

Factor out the constant 50 from the integral: \( 50 \int_0^{\infty} e^{-0.05 t} \, dt \). This simplifies the problem to finding \( \int_0^{\infty} e^{-0.05 t} \, dt \).

Step 4: Evaluate the Integral

The integral of \(e^{-kt}\) with respect to \(t\) is \(-\frac{1}{k}e^{-kt}\). Applying this, we get \(-\frac{1}{0.05} [e^{-0.05 t}]\) evaluated from 0 to \(\infty\).

Step 5: Apply Limits to the Integral

For the upper limit \(\infty\), as \(t\) approaches \(\infty\), \(e^{-0.05 t}\) approaches 0. So the expression becomes \(-\frac{1}{0.05} (0 - e^0) = \frac{1}{0.05} (1) \).

Step 6: Calculate the Total Output

Multiply the result from Step 5 by 50: \(50 \times \frac{1}{0.05} = 50 \times 20 = 1000\). Thus, the total output over the lifetime of the well is 1000 thousand barrels.

Key Concepts

Exponential Decay

Exponential decay is a fundamental concept in calculus and describes a process where a quantity decreases at a rate proportional to its current value. This is expressed mathematically by an exponential function, often with a negative exponent which causes the function to decrease over time. In the context of our oil well problem, the rate of oil production is represented by the function \(50e^{-0.05t}\). Here, \(e^{-0.05t}\) signifies exponential decay because the mathematical constant \(e\) (approximately 2.718) is raised to the power of \(-0.05t\), where \(t\) is in months.

• The base of the exponential function is \(e\).
• \(-0.05\) is the decay constant, indicating how quickly the production declines.
• The function will produce continuously decreasing outputs over time as \(t\) increases.

The concept of exponential decay is important in many fields, from natural sciences to economics, whenever diminishing processes over time are studied.

Improper Integral

An improper integral is a type of integral where either the interval of integration is infinite or the function has an infinite discontinuity. In this exercise, the integral that finds the total lifetime production of the oil well: \[\int_0^{\infty} 50 e^{-0.05t} \, dt\]is an improper integral because the upper limit of the integral is infinity.

• When approaching infinity, the function \(e^{-0.05t}\) decreases exponentially, which allows the integral to converge to a finite value.
• Finding a convergent value from an improper integral involves evaluating the limit as \(t\) approaches infinity.
• The steps include setting up the integral, simplifying it, and solving using limit properties and basic antiderivatives of exponential functions.

Understanding and calculating improper integrals is critical in real-world scenarios where predictions beyond certain finite limits are required, especially in economics and engineering.

Oil Production Calculation

Calculating the total oil output from the well requires integrating the rate of production from zero months to infinity. This is because, mathematical estimation can assume continuous operation for simplicity. The process starts from setting up an integral of the production rate:\[\int_0^{\infty} 50 e^{-0.05t} \, dt\]By factoring out constants, the integral reduces to:\[50 \int_0^{\infty} e^{-0.05t} \, dt\]Calculating this integral uses the antiderivative of the exponential function and the properties of limits:

• The antiderivative of \(e^{-0.05t}\) with respect to \(t\) is \(-\frac{1}{0.05} e^{-0.05t}\).
• When evaluated at the limits 0 and \(\infty\), \(e^{-0.05t}\) approaches \(0\) as \(t\) goes to infinity and equals \(1\) at \(t=0\).
• This results in the total calculation of \(50 \times 20 = 1000\) (thousand barrels).

Such calculations are invaluable for resource management, allowing stakeholders to strategically plan for resource utilization and shut down procedures.