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The rate of continuous money ilow starts at 5900 and increases exponentially at 2% per year for 20 years; Find the present value and final amount if interest eamed ...

Question

The rate of continuous money ilow starts at 5900 and increases exponentially at 2% per year for 20 years; Find the present value and final amount if interest eamed 490 compounded continuously:The present value is SL (Do not round until tho final answor Then round to tne noarest centas noeded )

The rate of continuous money ilow starts at 5900 and increases exponentially at 2% per year for 20 years; Find the present value and final amount if interest eamed 490 compounded continuously: The present value is SL (Do not round until tho final answor Then round to tne noarest centas noeded )



Answers

Money Flow The rate of a continuous flow of money starts at $\$ 5000$ and decreases exponentially at 1$\%$ per year for 8 years. Find the present value and final amount at an interest rate of 8$\%$ compounded continuously.

We have that the rate of continuous buddy for starts at one thousand dollars per year and it will increase exponentially at a rate of five percent per year or four years. And what we want to do is find the present value and the accumulated interest earned a three point five percent, which is going to be compounded continues. So to find accumulated, we need to find present value. So let's first, we'LL have a fine present value, so recall the present value is equal to zero to tea or capital. T is going to be our time interval with the function F thirty, which tells us our rate of money, which will bring even her here multiplied by E to the negative. Our little teeth beauty, where this little are supposed to be our raped, our interests that we're getting. So we're told that the money flow so F t is described by this part here, so it starts at one thousand dollars and an exponentially increases at five percent per year. So this time is that F A T is going to be so. It starts at one thousand and then there's going to be more supplied by E since it's exponentially increasing times zero point zero bye t. So this here will describe our value because, well, we plug in one into this. It would be one thousand times E to the zero point zero, which would be just that much more times a thousand. And so we have that. So we go ahead and plug this in to our present value. Christians were zero R time. Interval is going to be or so That's our capital. T f t. Like you said over there was gonna be one thousand five times e to the negative. So are are Is this three point by person here? So this is our little r. So this would be zero point zero zero zero points re five t and what we're going to do now Oh, drop that teeth right there. So I'm gonna first factor out of one thousand. I still have my interval from zero or and then I'm going to combine these two exponential tze and that will give what hold on our money one serious before. So this will be a being the zero point zero one five t and she might recall to integrate this is going to be e zero point zero one five t divided by its new power zero point zero one five and then we want to evaluate this from zero to four. So it's one thousand provided. So that's not really nice. Number two right down. But it will be too. So two thousand time for one thousand a thousand times, two hundred this one one, two, three, two hundred thousand divided by every. And then we want to evaluate this expression here at four injured. So when I funded for into the zero point zero one, our teeth end up with times, for this would be he to the point zero six and then subtract all to the zero, which was just being one. And so now this belly here will be approximately. Tell him he point zero six, which is about one point oh six. Subtract off one So the inside will be about point zero six and then we multiply that by two hundred thousand divided by three. So we end up about four thousand one hundred two points for war. So four thousand one hundred twenty dollars and four cents. And this here will be our who wasn't value and you might recall to find our accumulated. So, you know, emulated This here is equal to e to the r capital tee times our present value so e to the r capital t so capital tears or R is going point five percent. So this will be he to the zero point zero six and then p is the value just pounds approximately four thousand won twenty two point four four. And this here is approximately two more purpose. The last you had by you need to be point six, which will be about four thousand three hundred and seventy seven dollars and thirty five six. And so then this here will be our accumulated about.

Problem number 10 or off the is equal to $125,000 is equal to 20 years. K is equal to 6% so a is equal to integration from the root. ET are off team people came T d t, which is equal to integration from 0 to 20 $120,000 if our oh point or six t the bt. So the answer for this integration is equal to one toe. Five thousands over four point or six be our point. All 16 from the 20 which is equal to after substitution toe four million 833 576 ah.

Okay in this question that taxes given that is find the interest rate. So we have to find out our find the interest rate needed foreign investment of $4000 to double so PS $4000. An accumulated value double of the initial value. That is $8000. Okay. In five years. So t. Is also given five years. Okay when interest is compounded continuously. So we know when interest is compounded continuously. The formula for accumulated value is equal to P. E. Raised to the power artie. Okay. And now we have to find out the value of our hair rate of interest. So we will put all the values. So is 8000 equals two piece 4000 series to the power artie's areas to the power are and we know it is five and now we will divide both arctic questioned by 4000. So it will be 1000 divided by 4000 equals two years to the power five hours. And it will be two equals two years to the power five are or we can say it is to the power five R is equal to these are the same thing. And now we will solve this. So we will take natural logarithms on the bus are the equation. Because we have to solve this are there isn't the exponent? So Ln it is to the power five R is equal to L N two. And now L N A raise to the power be it will be B L N A. So it will be five R L N E equals two Ln two. And now natural algorithm has the best E. So Ln base E it will be one. So five R equals two Ln by two. And now we will divide both sides of the equation by five. So our will be won by five Ln two. And when we put the values in the calculator and the result will be approximately 0.1386 And in terms of percentage are will be approximately 13.86% per year or per annum. And this will be the final answer of this question. Thank you.

So here we're looking at continuous compound interest that's going to be the formula E. Equals E. To the RT. But the E. To the R. Key. And we know we're starting off with 30 500 and getting to 50,000. Um so we're starting at 35,000 action by 34:00. And then we're getting 50,000. And the rate at which this is happening is 4.75% compounded continuously. Mhm. There's a rape and we want to know the time now. So the way that we do this is we're going to divide here. Okay. And we're gonna take the natural log of both sides regarding this E. And then we'll divide by this value when we do that. We end up getting that the amount of time is going to be approximately 7.5 years.


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