Problem 21 - Applied Calculus

Question

An oil company discovered an oil reserve of 100 million barrels. For time \(t>0,\) in years, the company's extraction plan is a linear declining function of time as follows: $$ q(t)=a-b t $$ where \(q(t)\) is the rate of extraction of oil in millions of barrels per year at time \(t\) and \(b=0.1\) and \(a=10\) (a) How long does it take to exhaust the entire reserve? (b) The oil price is a constant \(\$ 20\) per barrel, the extraction cost per barrel is a constant \(\$ 10\), and the market interest rate is \(10 \%\) per year, compounded continuously. What is the present value of the company's profit?

Step-by-Step Solution

Answer

It takes 20 years to exhaust the reserve, with a present value of profits depending on the integral solution.

Step 1: Understanding the Extraction Function

We are given that the rate of extraction follows the function \(q(t) = a - bt\), where \(a = 10\) and \(b = 0.1\). This implies that the extraction begins at 10 million barrels per year and decreases linearly over time.

Step 2: Calculating Total Extraction Over Time

We need to find the time \(T\) when the total extracted oil equals the reserves. Use the equation for total extraction as an integral of the rate: \(\int_0^T (10 - 0.1t) \, dt = 100\).

Step 3: Solving the Integral for Total Extraction

Calculate \(\int_0^T (10 - 0.1t) \, dt = 10T - 0.05T^2\). Set this equal to 100 million barrels, giving the equation: \(10T - 0.05T^2 = 100\).

Step 4: Solving the Quadratic Equation

Re-write the equation as \(-0.05T^2 + 10T - 100 = 0\). Solve this quadratic equation using the quadratic formula: \(T = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\). Substituting \(a = -0.05\), \(b = 10\), and \(c = -100\), calculate \(T\).

Step 5: Calculating the Present Value of Profits

The profit per barrel is \\(10 (\\)20 - \$10). To find the present value of the profit, consider the integral of profit per year, adjusted for continuous discounting: \(PV = \int_0^T (10e^{-0.1t}(10 - 0.1t)) dt\).

Step 6: Solving the Integral for Present Value

Calculate the integral from the previous step: \(PV = \int_0^T [10(10 - 0.1t)e^{-0.1t}] \, dt\), which involves using integration by parts or substitution to solve.

Step 7: Evaluating the Expression

Solve the integral numerically or using software to find the present value of profits due to the continuous compounding interest. Ensure all constants and limits are correctly applied.

Key Concepts

Oil Extraction Model

The oil extraction model presented here is a mathematical representation of how oil is extracted from the reserves over time. The model uses a linear declining function to describe how the rate of oil extraction decreases each year.
This function is particularly useful as it simplifies the complex process into a manageable form that is easy to understand and calculate. The model assumes:

• An initial extraction rate of 10 million barrels per year.
• A constant rate of decline of 0.1 million barrels per year.

By understanding this model, we can predict how long the reserves will last and analyze the financial sustainability of the extraction plan.

Linear Declining Function

A linear declining function is used to model the decreasing rate of oil extraction. The function,\[ q(t) = a - bt \]where \(a\) and \(b\) are constants, describes a straight line with a negative slope. This indicates that as time \(t\) increases, the extraction rate \(q(t)\) decreases. Here, \(a = 10\) and \(b = 0.1\), leading to a downward linear slope.Key features of a linear declining function include:

• Predictability due to constant rate of decline.
• Simplicity in calculation and projection.

Using this function allows us to analyze when the oil reserves will be depleted by setting the integral of the function equal to the total reserves.

Present Value Calculation

Present value calculation is crucial in understanding the financial viability of oil extraction plans. It helps in determining the worth of future profits in today's terms by considering the time value of money.The formula to calculate the present value (PV) with continuous compounding:\[ PV = \int_0^T (10e^{-0.1t}(10 - 0.1t)) \, dt \]This involves:

• Calculating profit per barrel as \(10\).
• Using continuous compounding to discount future cash flows at a 10% interest rate.

Solving this integral gives the present value of expected profits from selling the extracted oil, adjusted for inflation and interest rates.

Quadratic Equation

To find out how long it takes to extract the entire 100 million barrels of oil, you'll need to solve a quadratic equation. This is derived from the integration of the linear declining function over time.The quadratic equation used is:\[ -0.05T^2 + 10T - 100 = 0 \]This equation emerges from integrating the extraction function and setting the total production to equal the reserve.The quadratic formula, \(T = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), is used here:

• Substitute \(a = -0.05\), \(b = 10\), \(c = -100\).
• Calculate to find \(T\), representing when the reserves are exhausted.

Solving this equation provides the time period necessary to deplete the oil reserves fully.

Continuous Compounding

Continuous compounding is a mathematical concept used to depict the accumulation of profits or interest over time continuously. Unlike simple or discrete compounding, continuous compounding assumes that the compounding process is happening at an infinite number of times per period.In the context of our problem:- The interest rate is 10% per year.- The formula modifies future cash flows for present value calculations using:\[ PV = \int_0^T (10e^{-0.1t}(10 - 0.1t)) \, dt \]

• The exponential function \(e^{-0.1t}\) reflects this continuous discounting, adjusting the profits in their present value.
• It helps in achieving a more accurate depiction of the present value of future earnings.

This approach ensures that the computed present value reflects the true economic value of future cash flows, taking into account the rigorous nature of continuous compounding.