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How long would it take money to double itself if it is earning interest at 11\% compounded (a) semiannually; (b) quarterly; (c) monthly; (d) daily; (e) continuously?

(a) $6 y 5 m 21 d$(b) $6 y 4 m 20 d$(c) $6 y 3 m 29 d$(d) $6 y 3 m 19 d$(e) $6 y 3 m 19 d$

Algebra

Chapter 4

Exponential and Logarithmic Functions

Section 6

Properties of Logarithmic Functions

McMaster University

University of Michigan - Ann Arbor

Idaho State University

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How long does it take mon…

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(a) How long does it take …

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Doubling Time How long wou…

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How long does it take for …

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(a) How long will it take …

Alright, This is kind of a fancy problem. Um, I really wish that they would just let what you leave the answer as a decimal. Then it would be more doable. But the main premise is that if we're doubling our money, what we want is the ending balance Instead of being a to B double p eso then when your first step to solve any problem, you divide both sides by P uh, both of these values of cancel out. So we're basically looking at this formula, Uh, and we know that the rate in the problem is 11 and the end is going to change depending on each problem. Eso How do we solve this without plugging in any ends is what you want to do is natural log both sides or log both sides. It doesn't matter. And, uh, what will change each time is is what Ennis eso We're looking at natural log of one plus 10.11 over and we'll instead writing to the anti power because we have the log, we can move that power in front. So what I'm saying is to solve for any tea in this problem, it's gonna equal the natural log of two divided by because all these air multiplication problems and natural log of one plus 10.11 over and again because it's 11% written as a desk one. So in part, A Uh huh. You semi annually, right? So my annually and equals two. So plug that in, um, And then I'll just go ahead and won't take me very long. It might just be a long video divided by, uh to natural law of one plus 10.11 divided by two. Actually, it might be smarter if I yeah, or to use. Ah. Anyway, when I do that, I get an answer of about 6.473 eso in part B. We're looking at quarterly. So that means that an equals four. Well, I could go to my calculator and just change. I'll do this. Read this. And in this end, to be four, I don't know if you can hear my calculator meet clicking away, so that gave me a T value. Oops. One such back to green to be about 6.388 Um, which should make sense. The more times you compound, the sooner your money will double you make more money. Uh, in part C, we are doing monthly so and equals 12 and in part d we're doing daily, which I'm going to assume 365 days in the year. I don't know if a teacher would say 365 and 1/4 but even that's technically not right. Um, and I don't want to get into an argument with anybody over that eso. I'm gonna go ahead and insert some 12 where I have a four. Oh, rats hit the wrong button. So this t value it was about 6.330 And this t value again. I'm not gonna do the 1/4 e just can't do three and 65. The difference won't be much anyway. 6.302 uh, and then impart e where we're compounding daily continuously. Excuse me. Um, it's really the same as the perp formula. A equals p e to the r t. The differences and it's still a equals two p. So it's the same premise is the other problem or the peas will cancel you. Natural log both sides. So natural log of two. That'll get rid of this e, uh, and then you divide by 0.11 The interest rate will tell you the t, so it's very easy. That's natural. Log of two divided by 20.11 and you'll get an answer of about 6.301 Which is really, really It's so minuscule that daily ah, lot of banks will do daily as well as continuously. Um, if you were to break that down two months, years, months and days, though, um, both of these answers air about six years. I already forgot what it was. Three months and 19 days. Okay. And the only difference for the previous ones is, you know, let her see, was 29 days so still, six years, three months? Um, and then the other ones, they're like, six years, four months, 20 days. So this be was four months, 20 days, So quite a big difference there. Which might be good if you're taking out alone. And then this one was five months, 21 days. Uh, still six years, which is pretty evident in the problem. You know, 6.6 point 6.6 years, six years, six years. It's a months and days that, you know, change very minuscule. E. After that,

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