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12.Marta invests bonus of S6000 in _ mulual (und that historically has retumed 129 year compounded quarlerly. Using (he compound inlerest formula A=p(i+Determlne ...

Question

12.Marta invests bonus of S6000 in _ mulual (und that historically has retumed 129 year compounded quarlerly. Using (he compound inlerest formula A=p(i+Determlne the account value In 6 years_

12.Marta invests bonus of S6000 in _ mulual (und that historically has retumed 129 year compounded quarlerly. Using (he compound inlerest formula A=p(i+ Determlne the account value In 6 years_



Answers

Find the future value, at $2.95 \%$ interest, compounded continuously for 6 years, of the continuous income stream with rate of flow $f(t)=2,000 e^{0.06 t}$.

And this problem going to take a look at, um, a rate of 5000 t e to the 0.1 T dollars per year that's invested forever at a rate of 6%. So are present value that we're trying to analyze. Here is the integral from zero to infinity of 5000 Tee hee hee to the 0.0 won t times e to the power of negative 0.6 t d t. And before we analyze this improper in a girl that simplifies we have 5000 t e to the negative 0.5 t d. T. The anti derivative of this one is gonna require integration by parts. So let's take a look at just this integral of t e to the negative 0.5 t. And, um, we'll deal with 5000 little bit later. So let's let you equal t, which gives us d'you equal d t and D be equal to e toothy negative through the power date of 0.5 t d t. And that gives us V equal to e to the negative 0.5 t divided by negative. 0.5 Okay. Our integration by parts formula then gives us the anti derivative um t multiplied by e to the negative 0.5 t divided by negative 0.5 minus. And then we're gonna multiply the times, do you? That ends up giving us a plus and then the inner girl of E to the negative 0.5 t divided by 0.5 times D t. And then our anti derivative of this. We'll get the tee times each and negatives your points, your five t and we divide that by negative 0.5 And then we do the anti derivative of this a second time, which gives us er minus e to the negative 0.5 t divided by 0.25 OK, so that's our anti derivative. And so let's use that now back into our improper interval. And this time we'll make sure we include 10,000 times the limit as t goes to infinity of this anti derivative we just found. And then it's gonna be evaluated from zero T. And so if you know if we let t go to infinity here, Uh, and we try to simplify this down. What we're gonna get is 5000 multiplied by one over 0.25 and that value in a dollar amount is $2 million.

The value of an investment that generates income at a rate of 5000 T. E. The 50000.1 T. Dollars per year forever Assuming an interest rate of 6%. So here's the formula for present value Big T. Is the time. And since we're doing forever that's going to be infinity. R. F. T. Is the dividend or which means how much the account pays every year. So this right here that's the dividend that's are empty and then little R. Is the interest rate. Okay so we have a zero to infinity. 5000 T. E. To the 50000.1 T. Times easier -160 D. T. All right. We have an infinity here which is not right. So we got to put limit as our goes to infinity. Well let's put the 5000 out in front to get it out of the way. 0 to R. T. Here we have E. To the 0.1 T. Times E. To the minus point oh 60. So you have those together and you could E to the minus point oh five T. You had those exponents together? D. T. All right I have to tease here. So it reminds me of the product rule. If I think of the product rule then I next thing of integration by parts. So that's what I'm gonna try. U. S. T. D. V. Each of the minus 0.5 T. D. T. If I take the derivative I get DT here. If I integrate Well I can't integrate unless I put a -105 which will put one out in front -1 over .5 integral E. To the U. D. U. So each of you. So now I have 5000 limit as our goes to infinity U v minus the integral which makes plus there VD you. Okay so now I'm going to integrate this last little piece, I'm gonna have to do that by putting in a -05 again. So uh -1 over .5 there. So I got 5000 limit. Our goes to infinity minus one over point oh five T E to the minus point oh five t -1 over .5 Squared E to the -105 T from zero R Uh -1 divided by .0520. Okay, so I have 5000 limit as our goes to infinity, I'm going to factor the one over E to the point oh five t out of both of those, So I get 20 t minus wait so far. Well just Plus 400 here. Yeah. Okay, they both had this E to the -105 T. Here it is. And then what was left was this? Which is -20 T. And then this which was minus .05 T. Looks like we're maybe you're going to have a minus sign problem. Oh no, no, we're not. Hopefully. All right. So now we got 5000 limit as our goes to infinity one over e. to the .05 are -20. -400 -1 over. Eat the zero which is one minus 20 time zero plus 400. Forgot to multiply by zero. There we go. So now I have 5000 infinity over infinity. If you think of this as the top and this is the bottom. So I'm gonna have to low petals rule that part are goes to infinity derivative of the top -20 derivative of the bottom. E to the minus Each at a .05 are Times .05 minus I got another minus sign mistake right there. That should be minus -1 times -420. All right. Take the limit as our goes to infinity, we got minus 20 on the top. We got E to the infinity on the bottom that goes to zero. So 5000 Time is 420 2,100,000 2,100,000 the end. Okay?

Okay, so for us to find our present value, given that we're calm pounding, um, by some amount, so that has an equation of P is equal to a times one, plus I to power of negative and where I is equal to our rate over are a number of compounding periods and and is equal to em. Times are number of years. Okay, so we're given that we have and amounts of 12,820 0.77 a rate of 4.8%. And we're compounding annually. So that's one time every year. And that's the power of one times six years. So plugging this into a calculator to find our present darling, he gets what he gets yet 9677 points. 13 So this was our principal or present value

So we're gonna become we're going to be computing some compound interest. Um Here I've just written the formula to find the amount of money made from compound interest after t amount of time. Um P. Is our initial investment ours our rate and is the amount of times were compounding and then tease our time limit. So here we were told that the initial investment was 500 so P. Is equal to 500. Our rate was equal to 6% and we're compounding annually. So that means we're gonna have one plus .6 to the power of tea since we're compounding annually. T is going to be in years. So this would be equal to 500 times 1.6. The power of teeth. All we want to do is figure out how much money are we going to have made after I'm six years. And we don't want to see like the profits from the compound. We just want to know the total amount of money that we have after six years. So we're just gonna plug in T. Is equal to six, we get 500 multiplied by 1.06 to the power of six. And so I'm just plugging this into my calculator really quickly we have 1.06. The power of six And then multiplied by 500 is equal to $709.26. So the amount of money we have after 66 years is $709.26 cents


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