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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.The analysis of tooth shrinkage by C. Loring Brace and colleagues at the University of Michigan's Museum of Anthropology indicates that human tooth size is continuing to decrease and that the evolutionary process has not yet come to a halt. In northern Europeans, for example, tooth size reduction now has a rate of $1 \%$ per 1000 years.a. If $t$ represents time in years and $y$ represents tooth size, use the condition that $y=0.99 y_{0}$ when $t=1000$ to find the value of $k$ in the equation $y=y_{0} e^{k t} .$ Then use this value of $k$ to answer the following questions.b. In about how many years will human teeth be $90 \%$ of their present size?c. What will be our descendants' tooth size 20,000 years from now (as a percentage of our present tooth size)?

(a) -0.00001(b) 10,536 years(c) $82 \%$

Calculus 1 / AB

Calculus 2 / BC

Chapter 7

Integrals and Transcendental Functions

Section 2

Exponential Change and Separable Differential Equations

Functions

Trig Integrals

Campbell University

Harvey Mudd College

University of Michigan - Ann Arbor

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this question. They talk about how our teeth are getting smaller, and in northern Europe it is at a rate of decreasing Eddery of 1% per 1000 years. So in the top left corner, I have that written as negative 1% over 100,000. So in the first part we have this equation here and they tell us we want to solve for Kay. So we know that when we saw for it in the way they have it set up, that we should end up getting this rate or K. But let's just see if that's the case or not. So first they give us this initial condition that I have 1000 is equal to 0.99 wiser. So let's go ahead and plug that in. So it should be zero point 99 Why zero is equal to y zero e to k. T. So we would want to divide each side by y zero now that be screwed all this over. So we want to first divide by y zero on each side so they cancel out, and then we would want to take the natural lock on each side to get that base E to cancel so we would end up with natural law of 0.99 is equal to. But I almost forgot to plug in the 1000 right here, or okay. I mean, poor teat. We'd have the 1000 right here. So we have 1000 okay? And then to solve. Okay, we're just going to bide each side by 1000. So we get that K is equal to the natural log of 0.99 over one 1000. So this looks slightly differently from the rate they give us a postcode. Approximate this to see it if it's in the same ballpark. So natural. Aga 0.99 is approximately negative. Zero point 0.1 over 1000 which is the same thing as our rate up here, because that percent is just 0.1 So we didn't really have to go through and Sulfur K. We could have just said it directly. But since they tell us to use that equation, that least help set it matches up. But so Kay is exactly this value right here, though. All right. Now they want to figure out When is our function equal to 90%? Or in other words, one is why equal to 0.9? Why not? So let's go ahead of just center created equal to them. 0.9. Why not is equal to you? E to the okay. We found in the last question. It should be natural log of 0.99 over 1000 and then we have tea. I don't have the wide not out front here. So again, we're gonna divide each side by white, huh? And then take the natural log that those cancel. And then we take the natural log this side as well. So we should end up with log a little bit better. We have a match log. As of the white knots, cancel on that side. The natural laws of 0.9 is equal to natural of of 0.99 over 1000 t. Now we would want to multiply excited by 1000 match plug of 0.99 On the right side, they cancel edit. That's gonna give us tea is equal to 1000 natural log of 0.9 all over match. Blawg of 0.9 nights. So whatever number this is Serbian years. So that's exactly But let's figure out this new. So one 1000 time's not too long. A 0.9 divided by natural log of 0.99 So this is about 10,000 483 years later that this will occur and depending on if you used exact values or if you rounded your answer may be slightly different from what we have here. But it should be somewhere in the ballpark of 10,000 years. 10,500 years. Yeah. So this is going to be our solution for part B. And now, lastly for the two sides after 20,000 years. Well, that means now we're gonna plug in 20,000 years into our equation. So that's gonna be why is equal to why not the Judy Kaye, which again is natural og of 0.99 divided by 1000 and then t is 20,000. So we're gonna plug in 20,000 years. And now let's just go ahead and see what gets wanted by all of this up. So 20,000 times the natural log of 0.99 divided by 1000. And then we're gonna exponentially ate that. So in the exponents, first, it should be something around approximately. Why not e raised to the negative 0.2. All right, so then exponentially ating. That should give us something around 0.8 two. Why not? So why not is our initial amount or how big our teeth are? So this implies. After 2000 years, they will be 82% of our current.

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