Question

An oil company discovered an oil reserve of 110 million barrels. for time >0, in years, the company's extraction plan is a linear declining function of time as follows: ()=20−0.1, where () is the rate of extraction of oil in millions of barrels per year at time . how long does it take to exhaust the entire reserve? time = years

Answer 1

To find out how long it will take to exhaust the oil reserve, integrate the rate of extraction from 0 to time t, then solve for t when the integral is equal to 110 million barrels. Performing the integration and solving the quadratic equation yields t = 20 years, meaning it will take 20 years to deplete the reserve.

Step-by-step explanation:

The question involves determining how long it will take for an oil company to exhaust an oil reserve through a linear declining extraction plan. The rate of extraction formula given is r(t) = 20 - 0.1t, where r(t) represents the rate of extraction in millions of barrels per year at time t in years. The reserve contains 110 million barrels.

To find the time t when the reserve will be exhausted, integrate the rate function from 0 to t and set the integral equal to the total reserve of 110 million barrels:

1. Setup the integral equation: ∫ r(t) dt = 110.
2. Perform the integration: ∫ (20 - 0.1t) dt = 110.
3. Solve for t: 20t - 0.05t^2 = 110.
4. Find the positive root of the quadratic equation: t = √{(20*110)*2/0.05}.
5. Calculate the value of t: t = 20 years.

Therefore, it will take the company 20 years to exhaust the entire oil reserve.