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The answers to most of the following exercises are in terms of logarithms and exponentials. A calculator can be helpful, enabling you to express the answers in decimal form.Suppose the amount of oil pumped from one of the canyon wells in Whittier, California, decreases at the continuous rate of $10 \%$ per year. When will the well's output fall to one fifth of its present value?

15.28 years

Calculus 1 / AB

Calculus 2 / BC

Chapter 7

Integrals and Transcendental Functions

Section 2

Exponential Change and Separable Differential Equations

Functions

Trig Integrals

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So they tell us. In Whittier, Calif. Warning of the amount of oil that is pumped from one of the canyon wolves decreases that a continuous rate 10% for you. We want to determine when will the Wells output Balto 1/5 of its present value. All right, so this sounds like an exponential because it's saying it's decreasing by 10% every single year. So we would write out that. So if Wyatt Tee is supposed to be how much oil we are pumping, there's gonna be Y t is equal. Why not e to the K? T? All right. And we want to first solve for Kay. And so they tell us it decreases by 10% her year. So we know why of one is going to be will decrease by 10%. Should be 0.9 of our original amount like this. And then we would divide by white, not meet side so they cancel out, and then we would take the natural log of 0.9, and I almost forgot to plug in Ah t here, who's this? Should have been one. And so then taking the natural log on each side gives us K is equal to the natural log of zero point. Not all right now they want us to solve for win is going to be one of us. Let's plug K in First was gonna be Why not? Times e to the t natural log of zero point signed. And we want this to be equal to one here. Why not? So again, the why nots we're just gonna cancel each other and we would take natural log on each side to get the natural log of one is equal to t natural off of zero point. Now, we would want to go ahead and divide each side by natural light of 0.9. So that's gonna give us t is equal to the natural log of 1/5 which is 0.2 over natural log of 0.9. And let's see, this is going to be approximately natural. Aga 0.2 is about 1.6 and then divided that by natural aga 0.9 this gives about 15.3 years. So after about this time, well, we have got to 1/5 of be production

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