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:

where \(q\left( t \right)\) is the rate of extraction of oil in million of barrels per year at time t, and a = 10, b = 0.1.

(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?