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. 


c. Find the minimum decay constant k for which the total oil reserves will last forever.

Verified step by step guidance

1. Understand the problem: The extraction rate is given by the function \(r(t) = r_0 e^{-kt}\), where \(r_0 = 10^7\) barrels/year and \(k > 0\) is the decay constant. The total oil reserve is \$2 \times 10^9\( barrels. We want to find the minimum value of \)k$ such that the total extracted oil over infinite time does not exceed the reserve.
2. Set up the integral for total oil extracted over time from \(t=0\) to \(t=\infty\): \(\displaystyle \int_0^{\infty} r(t) \, dt = \int_0^{\infty} r_0 e^{-kt} \, dt\).
3. Evaluate the integral: Since \(r_0\) is constant, factor it out: \(r_0 \int_0^{\infty} e^{-kt} \, dt\). The integral of \(e^{-kt}\) from 0 to infinity is \(\frac{1}{k}\), so the total extracted oil is \(\frac{r_0}{k}\).
4. Apply the condition that the total extracted oil must be less than or equal to the total reserve: \(\frac{r_0}{k} \leq 2 \times 10^9\) barrels.
5. Solve the inequality for \(k\): Rearranging gives \(k \geq \frac{r_0}{2 \times 10^9}\). Substitute \(r_0 = 10^7\) to find the minimum decay constant \(k\).

Key Concepts

Exponential Decay Function

An exponential decay function models quantities that decrease at a rate proportional to their current value. Here, r(t) = r0e^(-kt) represents the rate of oil extraction decreasing over time, where k > 0 controls how fast the rate declines. Understanding this helps analyze how extraction slows and affects total resource depletion.

Definite Integral and Total Quantity Extracted

The total amount of oil extracted over time is found by integrating the rate function r(t) from zero to infinity. This integral sums all infinitesimal extractions, allowing us to compare the total extracted oil to the reserve limit. Mastery of definite integrals is essential to solve for conditions on k.

Convergence of Improper Integrals

Since the extraction rate decreases exponentially, the integral from zero to infinity is an improper integral. Determining whether this integral converges (i.e., has a finite value) is crucial to ensure the total extracted oil does not exceed the reserve. This concept helps find the minimum decay constant k for sustainable extraction.