An oil company discovered an oil reserve of 100 million...

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?

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?

Key Concepts

Linear Decline Functions

A linear decline function models a quantity that decreases at a constant rate over time. In contexts such as resource extraction, it represents scenarios where the extraction rate starts at a certain amount and diminishes steadily, allowing planners to forecast when the resource will be exhausted.

Definite Integration

Definite integration is the process of calculating the total accumulation of a continuously changing quantity over a set interval. It is used to determine quantities such as the total extraction over time when the rate of extraction is given as a function of time.

Net Present Value (NPV)

Net Present Value is a financial metric that sums the current values of future cash flows, subtracting any initial expenditures. In resource extraction, NPV is used to evaluate the profitability of a project by accounting for both the timing of revenues and costs as well as the impact of the discount rate.

Continuous Compounding

Continuous compounding refers to the mathematical model in which interest accrues instantly at every moment, rather than at discrete intervals. This concept is essential in finance for accurately discounting future cash flows to their present values, especially when dealing with continuous cash flows.

Discounted Cash Flow Analysis

Discounted Cash Flow (DCF) analysis is a valuation method used to estimate the value of an investment based on its expected future cash flows. By applying a discount rate to future revenues and costs, DCF analysis accounts for the time value of money, thereby determining the present profitability of an investment.

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