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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 that in any given year the number of cases can be reduced by $25 \%$ instead of $20 \%$a. How long will it take to reduce the number of cases to $1000 ?$b. How long will it take to eradicate the disease-that is, reduce the number of cases to less than $1 ?$

(a) 8 years(b) 32.02 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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University of Michigan - Ann Arbor

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So this is a continuation of example, or and in that they talk about how many cases can be reduced over the course of ah year, and in that problem, we assume 20%. Now They're telling us to assume our rate is 25% after a year. And what we want to determine is how long will it take to reduce the number of cases to just 2000 as well as how long will it take to eradicate the disease, which they tell us it's the same thing is finding the cases is less than one. All right, so let's go ahead and use everything that we know so far and let's soul for Kay first. So we're changing the 25% reduction. So that means after a year, which means after one year, why of one should be 75% of our initial, which is going to be this 10,000 to remember. This is our initial value. Maybe not. But why not? So this is going to be 0.75 oh, 10,000 than E to K is equal to. Then we have 10,000 on the other side. He to the Katie. Now, the only reason why I'm not going to multiply the left hand side is because now notice if we were to divide each side by 10,000 the 10,000 just cancel out so we could take natural log. And I need to plug in one. Her tea right here, Actually. And that we would take natural logs, would get natural log of 0.7. Bye. Busy too. Okay, so that's going to be our K for this. So we can write out our equation now to be that why of tea is equal to 10,000 times E to the tea time statue log of 0.5? No, no. So we can actually pull that tea inside using the power rule or logs. And it would be natural elegance. 0.75 race, two teeth. And then these camps out. And we're just gonna be left with 10,000 times 0.75 raised the tea. And I'm just gonna write like this because I feel like it'll helps simplify our calculations a little bit. Actually, let me go ahead and move on this toothy side a little bit. That way we have some room toe work on parts A B now for part A. They wanted to figure out when this is equal to 1000. So we're gonna have 1000 is equal to 10,000 times 0.7 by two teeth. And so we would divide each side by 10,000 and that's just going to give us 12 is equal to 0.75 t activity. Then we can take the natural log on each side. So reviving the left side that negative match log of 10 is even too tee times the natural log of 0.75 Now we could divide each side by natural aga 0.7 by to get that t is equal to negative natural law 10 over natural law of zero points and five And then let's see what value this gives us about negative natural log. Uh, 10 is about negative. 2.3. Divide that by natural of 0.7. Bye. So this gives about eight years. So this is our solution for part a. Now, how about part be well, they want us to find when this is going to be less than one so we're gonna have one less than 10,000 time 0.75 Racing team member's gonna follow the same steps that we did before, so divide over by 10,000 won over 10,000. Then we would want to take the natural log on each side as well. It would be natural log. Oh, tee times the natural log of 0.75 Now, remember, this is a negative number. So if we were to divide each side Bye the natural log of 0.75 Remember, we're gonna have to flip our, uh, inequality here. So let's write that over here, though. So again, I'm going to rewrite that top one as the negative natural log of 10,000 that we have the natural log of 0.75 And now, like I was saying, we flip our inequality here. Actually, I mean, not right equal. We live big witless to get that and so on the right side, we would just have to slips, estimate what this value is. So we have negative natural log of 10,000 divided by the natural log of 0.7 I. And so this is about 32. So we will get when t is larger than 32 years, this will be eradicated. So let me just right over here. So 32 years about for part B?

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