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Problem 2: Problem 25Previous ProblemProblem ListNext Problempoint)A continuous annuity with withdrawal rate N = 81,200/year and interest rate r = 4% is funded by a...

Question

Problem 2: Problem 25Previous ProblemProblem ListNext Problempoint)A continuous annuity with withdrawal rate N = 81,200/year and interest rate r = 4% is funded by an initial deposit Po _ (a) When will the annuity run out of funds if Po 829,000?The annuity runs out after approximately Answer to the nearest whole year:years_Which initial deposit Po yields a constant balance? Po

Problem 2: Problem 25 Previous Problem Problem List Next Problem point) A continuous annuity with withdrawal rate N = 81,200/year and interest rate r = 4% is funded by an initial deposit Po _ (a) When will the annuity run out of funds if Po 829,000? The annuity runs out after approximately Answer to the nearest whole year: years_ Which initial deposit Po yields a constant balance? Po



Answers

A continuous annuity with withdrawal rate $N=\$ 5000$ lyear and
interest rate $r=5 \%$ is funded by an initial deposit of $P_{0}=\$ 50,000 .$
\begin{equation}\begin{array}{l}{\text { (a) What is the balance in the annuity after } 10 \text { years? }} \\ {\text { (b) When will the annuity run out of funds? }}\end{array}\end{equation}

In this problem, we're told that we need to put forties in honor. Circular hearing? No, but this problem we have to use concept Circular permutations know what is interesting about a circular communication, as opposed you may be a linear formulation is that in a circular permutation, usually clockwise and counterclockwise arrangements are considered as the team arrangement. As a result from the concept of circular permutation, the formless is that if we need to arrange end things in a circular order than the number of ways off doing for ending en element around the circle, distinctly the bomb limos, gripper mutations is the number of reasons, in fact, or yell upon end, which can build on us and minus one Victorian. Now, using this formula here if we need to arrange bordering. So for four keys to be a realtors are clearing. The number of ways will be and factorial upon end, that is, four factorial upon four gives us 24 upon four. That is so. There are only six distinct ways in which we can arrange Porky's around a circular Keating

In this problem. We're told that someone's going to deposit $2000 into retirement and it's going to be in there for 45 years. So and they're being paid 12% interest. That's going to get compound it annually. We want to know what is the amount of the annuity at the end of the 45th year? Well, we can do this by thinking about, uh, coming up with a series that in this case you're gonna find is going to be geometric. But it's gonna be a little bit backwards because you usually think what you initially deposit would be your ace of one term. But we're gonna actually think about it in terms of, well, what would be, ah, what is your account making at the end? So at the end of the 45th year, when you deposit $2000 that's only going to earn interest for that one a year so we can multiply this to figure out how much interest is he'll earn is going to be 1.12 And the reason why that base is 1.12 is because remember our rate as a decibel. So 12% is a decimal would be 120.12. And because we're adding it on to the amount that we've already deposited, we need to do one plus 0.1 12. That's where the 1.12 comes from. So this 2000 times 1.12 is gonna represent how much money he's gonna earn in that very last year. Well, the year before the last year again, he's gonna deposit $2000 But he's gonna earn this interest over two years of time. So we would multiply that by 1.1 to race to the second power, because again, that's going to represent how much interest he earns after two years. So, for example, the third year before he would earn 2000 times 1.1 to race to the third power. And this is gonna keep going until now. We have to think of it as the very first year we put money in. Well, I'm just gonna sneak in and down here. We're gonna deposit $2000 but he's gonna earn interest on this $2000 for 45 years, so we're gonna multiply it by 1.1 to raise to the 45th Power. So now remember, we're trying to find the amount of his annuity, which is essentially the some off this Siri's will remember to find the sum for a finding geometric. Siri's It's s a Ben is equal to a sea of one times to quantity of one minus our race to the end power. And it's all getting divided by one minus. R. Well, in this particular case, we know end and it's the time. Well, Theresa was 45 years, so we're trying to find us so 45 equals to a someone which we said was 2000 times 1.12 Then we need to multiply that by one minus R, which is 1.1 to race to the end power, which is the 45th. Now we need to divide this by one minus R, which remember is 1.12 So now we're just gonna have to go ahead and simplify this expression. So when we do this, what we're going to do first is I'm going to multiply the two terms outside my prep, seeing the new murder, so I'm going to do 2000 times 1.12 which is equal to 2240. So I have es of 45 is equal to 2240. And then I would take the value inside my parentheses, which is one minus 1.12 raised to the 45th Power. And what you'll find is you get negative 1 62.9876039 and it's all getting divided by one minus 1.12 which is equal to negative 0.0 point 12 So now all we have to dio is multiply than values in the numerator and then divide it by our denominator. So when we multiply the 2240 by the negative 162.9876039 and then we take that answer and we divide it by negative 0.12 What you're going to find is you're going to get, um, excuse me. Uh, three three million. Lets put the dollar sign here three million 42,000 435. And if he rounds in nearest sent 27 cents. So that's how much annuity this person would. Oh, with earn over 45

All right in this situation there giving us that we're investing $5,000 at a rate of 4%. And it's saying that it's compounded continuously and we want to know how much money is in our account after eight years. Alright, anytime they tell us that our our problem is being compounded continuously, we are going to use the A. Is equal to P. Which is our principle E. To the R. T. Formula instead. So anytime it says compounded continuously, I actually find these problems easier. All right. You do want to change your rate um into a into a decimal and set of a percent. So 4% is for over 100 Which is .04. All right. But now we can just plug into this formula. Use a calculator and get our solution. Okay. So we're solving that our A. Is equal to R. P. Is 5000 E. Our rate is .04. And we are solving this for eight years. You can type this into a calculator exactly how it's written. You just got to make sure you find the E button. Uh You can multiply the point oh four out and ate out if you wanted to. First. It's up to you. All right. When I do type this into my calculator, I get that We will have 6000 $885 and 64 cents in our account

They probably have a continuously compounded interest problem. Continuously compounded interest problem is what I call the park Formula days equal P. E. To the RT. Now P is your principal. That's just how much you're going to invest mm Or is the rate and then t is the time. And notice I didn't write the letter E. Here. That's because he is just the constant. Okay. It's the number 2.71 and so on. And so it will be the same every time. Now here on this problem we are investing $5000. So this is 5000 times E. So and then the interest rate is .04. This is .04 And we're going to do this for eight years .04 times eight. Now from here with news type this indirectly. Like what's on a calculator. All right. And they should give us 6000 $885 And 64 cents.


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