Depletion of natural resources Suppose r(t) = r0e^−kt, with k&...

Question

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.

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

Accepted Answer

Welcome back everyone to another video. A country's coal extraction rate is modeled by S T equals S 0 E to the power of negative MT, where M is greater than 0, and S subscript 0 is equal to 5 multiplied by the power of 6 tons per year is the current extraction rate. Finds of T. The total amount of coal extracted after 2 years. So for this problem, S of T is going to be the integral of the rate function, right? We're going to integrate lower cases of T with respect to T. From 0, which is our initial time right up to T, where T is some time. After the initial time T equals 0, we're going to identify our functions of T this way. So what we're going to do is simply substitute for the function. We know that S of T is equal to S subscript 0 multiplied by E raises the power of negative M T. We're going to integrate that from 0 to T. We can factor out as subscript 0 because it's a constant. And multiply it by the integral of Ease the power of negative MTDT from 0 up to T. Let's go ahead and integrate. We are going to write S subscript here, which is our constant. The integral of E raises the power of negative MT is going to be E raises the power of negative MT, and according to the reverse chain rule, we're going to divide by The slope of the linear term, right, because our exponent consists of a linear function, so divide negative. We We can also write it as negative, as subscript 0 divided by M multiplied by E raises the power of negative M T evaluated from 0 to T. And now let's apply the evaluation theorem. We have our constant negative. MT, so we're substituting T for T, meaning nothing changes, and we're subtracting E. Raise the power of negative M multiplied by 0, right? So, essentially here we are subtracting 1 because E raises the power of 0 is 1. And now what we're going to do is simply rewrite it and Go back. And swap the signs with currencies minus E raised the power of negative M T. Our final step is to replace S subscript here with the actual value, which is 5 multiplied by 10 to the 6th. And we're dividing that by M. M is a constant, we know that it's positive, and we're multiplying by 1 minus E raises the power of negative MT, which is our final answer for this problem. Thank you for watching.

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