Depletion of natural resources Suppose r(t) = r0e^−kt, with k>0, is the rate at which a nation extracts oil, where r0=10⁷ barrels/yr is the current rate of extraction. Suppose also that the estimate of the total oil reserve is 2×10⁹ barrels. 


a. Find Q(t), the total amount of oil extracted by the nation after t years.

Verified step by step guidance

1. Identify the given rate of extraction function: \(r(t) = r_0 e^{-kt}\), where \(r_0 = 10^7\) barrels/year and \(k > 0\).
2. Recall that \(Q(t)\), the total amount of oil extracted after \(t\) years, is the integral of the rate function \(r(t)\) from 0 to \(t\): \(Q(t) = \int_0^t r(\tau) \, d\tau\).
3. Substitute the given rate function into the integral: \(Q(t) = \int_0^t r_0 e^{-k\tau} \, d\tau\).
4. Integrate the exponential function with respect to \(\tau\): \(\int e^{-k\tau} \, d\tau = -\frac{1}{k} e^{-k\tau} + C\).
5. Apply the definite integral limits from 0 to \(t\) and multiply by \(r_0\): \(Q(t) = r_0 \left[-\frac{1}{k} e^{-k\tau} \right]_0^t = r_0 \left(-\frac{1}{k} e^{-kt} + \frac{1}{k} e^{0} \right)\).

Key Concepts

Exponential Decay Function

An exponential decay function models quantities that decrease at a rate proportional to their current value. In this problem, the extraction rate r(t) = r0e^(-kt) shows how the oil extraction rate decreases over time, with k > 0 controlling the speed of decay.

Definite Integral for Accumulated Quantity

To find the total amount extracted over time, we integrate the rate function r(t) from 0 to t. The definite integral sums the instantaneous extraction rates, giving the cumulative quantity Q(t) extracted after t years.

Initial Conditions and Parameters

Understanding the given constants, such as the initial extraction rate r0 and the total reserve, is crucial. These parameters set the scale and constraints for the problem, ensuring the solution is physically meaningful and consistent with the resource limits.