Depletion of natural resources Suppose that r(t)=r0 e^-k t...

Depletion of natural resources Suppose that $r(t)=r_{0} e^{-k t}$ with $k \geq 0,$ is the rate at which a nation extracts oil, where $r_{0}=10^{7}$ barrels / yr is the current rate of extraction. Suppose also that the estimate of the total oil reserve is $2 \times 10^{9}$ barrels. a. Find $Q(t),$ the total amount of oil extracted by the nation after $t$ years. b. Evaluate $\lim _{t \rightarrow \infty} Q(t)$ and explain the meaning of this limit. c. Find the minimum decay constant $k$ for which the total oil reserves will last forever. d. Suppose $r_{0}=2 \times 10^{7}$ barrels/yr and the decay constant $k$ is the minimum value found in part (c). How long will the total oil reserves last?

Depletion of natural resources Suppose that $r(t)=r_{0} e^{-k t}$ with $k \geq 0,$ is the rate at which a nation extracts oil, where $r_{0}=10^{7}$ barrels / yr is the current rate of extraction. Suppose also that the estimate of the total oil reserve is $2 \times 10^{9}$ barrels.
a. Find $Q(t),$ the total amount of oil extracted by the nation after $t$ years.
b. Evaluate $\lim _{t \rightarrow \infty} Q(t)$ and explain the meaning of this limit.
c. Find the minimum decay constant $k$ for which the total oil reserves will last forever.
d. Suppose $r_{0}=2 \times 10^{7}$ barrels/yr and the decay constant $k$ is the minimum value found in part (c). How long will the total oil reserves last?

Key Concepts

Exponential Decay

Exponential decay refers to a process where a quantity decreases at a rate proportional to its current value. In a variety of contexts, such as resource extraction, this decay is modeled by a function that multiplies an initial amount by a decaying exponential term, which captures the gradual reduction in the extraction rate over time.

Integration of Exponential Functions

Integrating an exponential function is a fundamental technique for calculating cumulative quantities over time. When dealing with rates that decay exponentially, the integral provides the total amount accumulated, which is especially useful when determining how much of a resource has been extracted by a specified time.

Improper Integrals and Limits

Improper integrals are used when integrating over an infinite interval or a region where the function behaves unboundedly. In this context, evaluating the limit of the integral as time approaches infinity reveals the asymptotic behavior of the total accumulated extraction, providing insights into whether the cumulative extraction converges to a finite value.

Sustainability and Parameter Analysis

Analyzing parameter conditions, such as the threshold decay constant, is crucial in determining long-term sustainability of resource use. By identifying the minimum decay constant that ensures the cumulative extraction does not exceed the total reserve, one evaluates the balance between extraction rates and resource longevity, a key concept in resource management and sustainability studies.

Transcript

00:01 Okay, here we have a problem of the depletion of natural resources.

00:05 We have r of t, which is the rate in which a nation extracts oil, and we have 10 to the power of 7.

00:11 The actual problem is actually going to have r0e to the power of negative k, t.

00:17 But i'm replacing up 107 because r0 is equal to 10, 7 for our purposes.

00:22 10 to the power of 7 for our purposes.

00:24 We also have to assume that we have, in the reserves of oil, 2 times 10 to the power of 9.

00:32 Or 2 billion.

00:34 So we're going to find qt.

00:35 To find qt, we are going to just find the integral for this.

00:40 So if we find the antis derivative of our rate, find the total oil consumed or extracted, we get 10, 7, we get 1 minus e negative kt over k.

01:02 The reason we have the k in the denominator here is because we have the k here.

01:07 So that's what we get from that.

01:10 Now, what it asks us to do next, it asks us to figure out, if we have t to the power of infinity, what do we get? so, e to the power of infinity, or sorry, t to the power of infinity.

01:26 If we have t to the power of infinity, this part here goes to zero.

01:34 Because negative infinity, e to the power of negative infinity, becomes 1 over e to the power of infinity.

01:39 And if one is being divided by something infinitesimal, it approaches zero.

01:45 With that, we get 10 to the part of 7 times 1 over k.

01:52 So the total amounts of barrels of oil extracted over an infinite item of time, we get 10 to the power of 7 over k.

02:00 That's the meaning of the limit.

02:03 So that means the nation has to have at least this much oil unreserved...

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