a-if-3000-is-invested-al-5-interest-find-the-value-of-the-investment-at-the-end-of-5-years-if-the-in

Question

(a) If 3000 is invested al $ 5\% $ interest, find the value of the investment at the end of 5 years if the interest is compounded
 (i) annually, (ii) semiannually, (iii) monthly, (iv) weekly, (v) daily, and (vi) continuously.
(b) If $ A(t) $ is the amount of the investment at time $ t $ for the case of continuous compounding, write a differential equation and an initial condition satisfied by $ A(t). $

Wen Zheng · Accepted Answer

The problem is, if three South End is blasted at our present interest, finding the while you have the investment at end of five years. If the interest is compounded one annually, semi annually. Masterly quickly daily. Andi continuously So one three. Southern Ham&#x27;s one plus. Is there a point? You&#x27;re far too far. This is about three eight you made on page four two. Three thousand one. Class zero point zero five over two to the power of ten. Statistics about three eight. Both their old aren&#x27;t too far on three, three thousand times one plus Pierrepont It&#x27;s your father. Wow, She&#x27;s our of sixty. This is about three feet off. Zero point Sarah wait or No. Three Solvent Ham&#x27;s one Plastic zero point zero five two fifty two Time The power half love times two This is about three Ain&#x27;t off one one two six one Love three thousand That&#x27;s one plus zero point two zero five over three six five to the power of five times three six five. This is about three eight five two on two one zero six three thousand halves each is the power of syrup on zero five. Hi. This is about create off two point zero eat P if eighteen amount of investment at Tom T. But kids of continuous compounding, right? A differential equation on DH. Unusual conditions. Satisfied? I hate you. So have they. Yeah, is the cunt you are? A On initial condition, the zero is equal to three thousand.

Cyrielle Lorio · Answer

Okay, So what we have here is a problem involving a money in a bank account That is compounded. Continue with me at 5% interest rate or 0.5 So money is also deposited at the cantina. Was cash flow amounting to $1200 per year. So we are required together V off the or not theory, let there be the particular solution. Even an initial contribution re of 060 and the money in the account off their five years. Okay, so the good be ISS, we can ram percent the money in the bank account as they be over the B is equal to 0.5 b plus 1200. We can barter isolate the 0.5 and you can represent the equation as 0.5 times be lost. But before 1000. So this is now a separable differential equation on, and we can rearrange our equation as the B over B plus 24,000 is equal to 0.5 bp, so integrating both sides will have, and in off B plus 24,000 is equal to zero point certified B Let&#x27;s see, so we can get the power off the of both sides and we do this by doing here is to Ln off being plus 24,000 is equal the eeriest 0.5 b. Let&#x27;s see so we can represent the left side US B plus 24,000 And remember that See, here is just a concept so it can think any value so we can write the right side. Us. CE raised to 0.5 p. So the further isolate the B on the left side. But then I started 24,000 r right side. So we&#x27;ll have b is equal to see raised to 0.5 b minus 24,000. So this is our general solution on, but they want again. It&#x27;s the particular solution for letter V. So for B of zero is equal to zero who just substitute this over equation and we&#x27;ll have zero is equal to C E races, you know, 00.5 times zero my nose 24 out sent. So we know that this will just be equal to one and we&#x27;ll have si is equal to going before said he started it. I think Arctic we should well have be off the Is Ecuador going before 1000 ce raised 0.5 b minus going before thou sent. So this is our particular. So they shot. Okay, so for another See, we want to find the amount off the money in the bank at these equal five? Nope. Sorry. So for leather, see, at the in support of five, we just substitute this to our particular solution And we have 24,000. Well, sorry. There should not be any See here. So our particular solution is done before 1000 erased 0.5 B minus with before 1000 G. So be off. Five will be equal to 24,000 e raise to 0.5 times five miners. 24,000. So computing for this this is approximately equal Cool. $6816.61. So this is our final answer

Cyrielle Lorio · Answer

Okay, So what we have here is ah, money. The posse that it back account with an initial deposit of 5000 physical to zero and compounded continuously at our interest rate of 1.5% or 0.15 So we are required to gets its differential equation on and then so far, and then find the amount of money at P is equal to 10. He&#x27;s our solution. I Sorry about that. Okay, So for leather A, we can express BB over B P is equal to 0.15 b. So this is our answer in leather. Easy. Now we&#x27;re asked the soul for its solutions. Suffer be. What we want to do now is the won&#x27;t apply both sides by one over. Be so by doing that we have BB over B B B is equal to 0.15 Now we want to multiply both sides by BT. By doing that, we now have baby Overbey. Is it about 0.15 b d. So integrating both sides are answer Would be Eleanor be is equal to 0.15 The let&#x27;s see So we want the race each side to e and by doing that will have B is equal to erase to 0.15 thing. Let&#x27;s see. So remember that See, here is just a constant so it can take up any value. Therefore, we can express this. A creationist B is equal to see a raise to 0.15 b. So this is our general seusshow. So we want to find its particular solution. So we do this by finding the value of C. And we do that by taking use off the, um initial by new initial conditions given from the problem. So, uh, be is it about zero? B is equal to 5002. Substituting these to our general solution we have 5000 is equal to see erased 0.15 90 So this will just be equal toe one. Therefore, the value of our constant C is equal to fight out son rewriting dollar General Sougou Sean, our particular solution would be be is equal to 5000 e raise the 0.15 feet the business, our particular solution and we make yourself these The answer that there. See? So for leather, See at the physic. Well, then, what is the value of me? So we just substitute the give and be to the equation and our solution would be being is equal to 5000 Erased 0.15 Thanks, then, which is approximately equal to $5809.17. So this is our final answer.

Caleb Miller · Answer

Hey, guys, welcome back. We&#x27;re told that we have initial investment of $3000 at 5% interest. We were asked to figure out how much said investment will grow at different interest annual interest. Semiannual monthly, weekly daily. Continuously. We&#x27;ll use our equation. Our amount after sometime is able to our initial amount our initial amount being the 3000 plus our rate over end over end to the team and r T. It&#x27;s how many years. So in this case, he would be five for all these things two years ago to five. And then end is how many times per year times per year. So first we were asked to figure out what happens. It&#x27;s calm, pounded annually, so his compound annually than end is simply one part a and his one I have 3000 one plus r ever end are is 0.5 Number one raised to the end T and we said was one t value is five. And if you do the math, that comes out to 3000 828 0.84 Next problem. Rest figure. What happens if this compound id semi annually? So for semi annually than end will be equal to two. I will say that a city should have said That appears well. A city is equal to 3000 one plus 10.5 and is not to times that by to to the five. And that will give us a value of 3000 140 0.25 Next, we&#x27;re asked to figure out what happens if it&#x27;s calm. Pounded monthly 12 months and years and his 12 and we have a He is evil Toe 3000 one plus 10.0 5/12 0 5/12 I replied by two times 12 and that comes out to 3000 850 0.0 eight. Next to ask to figure out what happens if it&#x27;s compound ID weekly in its weekly. There are 52 weeks in a year, so our equation becomes. Eight of the tea is equal to 3000 3000 one plus 10.5 over 52 multiplied by 52. I&#x27;m sorry. I apologize up here. I had a slight mistake. Our experts should have been 12 multiplied by five and this answer is still correct. That was the correct answer. Going back now. Down to Part D. You do this math? Get a value of 3000 851 0.61 now for part, you were asked for your what happened to his compound it daily. There are 365 days in a year. To me, it&#x27;s not only beer, we&#x27;ll have a Sebti people of 3000 one plus 0.5 3 65 raised to the five years times 365 times every year. If you do the math on that one, that comes out to 3000 800 3000 800 52 $01. Part of the rest of you know what happens. It&#x27;s compound to continuously. Basically, we&#x27;re saying what happens when N goes to infinity or any ghost Infinity, Our equation turns into a city is ableto our initial value. Times e. Raised to the rt becomes 3000 times E Raised to the are just 0.5 times T just five, and that comes out to three 1000 800 to point a sense. Now it&#x27;s part F that includes parts A through F, and that includes the problem. Thanks for watching

Ankit Gupta · Answer

Ocean says that if $10,000 is invested at an interest rate off, 3% bloodier, compounded semi annually, find the value of investment after the given number off years. So the formula for finding this amount is equipped to be in tow. One plus R divided by entries but empty their peace. The principal amount are is the date and decimal, and it&#x27;s the number of times interest is compounded, buddy it and he is the number off years. According to a question. He&#x27;s a quick 10,000. Art is equal to three person, so it would be 0.3 since the interest is being compounded semi annually and will be equal to don&#x27;t. So we are given the value off key and we have to find uncertain what each value So in part eighties 1/4 to 5 years. Therefore able be quick, dude, 5000. Ah, sorry. This is 10,000 years. 10,000 into one place, 0.3 Do right by two days. Part goingto five my Justin. So therefore this amount is equal to 11,605 $0.4 in part B is equal to 10 years. So even be quick. 10,000 into one place. So 0.3 divide by two Jace, but 20. So this is coming out to be 13,468 $0.55 and biopsy time. Peter, this 15 years, therefore able big were to 10,000 into one plus 0.0 treat divide by two, they spot 30 so this is coming out to be 15,630 $0.8.

Bobby Barnes · Answer

they tell us that formula A of T is equal to 1000 times 1.5 Raciti will be the amount of money we will have after two years. If we invested $1000 into account, that phase pays 5% interest that compounds annually. So they want to find the average rate of change over the first year or 1st 5 years and the 2nd 5 years and then they wants to estimate the instantaneous rate of change at five years. So let&#x27;s go ahead and start with the 1st 5 years, which is gonna be 05 So we&#x27;re gonna plug everything into that average rate of change equation only difference says instead of f, we&#x27;re going to use a There&#x27;s gonna be a of five minus a zero all over five minus zero. So let&#x27;s look that in. So we have 1000 times 1.5 raise to the minus 1000 and times 1.50 which is just gonna be one. And then we would divide this by five. So first knows we could divide this five and each of these terms such as could be 200 so we need to find. So 1.5 raised to the fifth times 200. So that&#x27;s about 255 minus 200. So this is somewhere in the ballpark of 50 five dollars and 26. And so this is how much interest we would earn on that 1st 5 years. Now, what about the 2nd 5 years? Well, now we&#x27;re going to do a of 10 minus a five, Then be 10 minus five. So we just plug it again again. So it be 1000 1.5 race and 10th minus 1000 1.5 raised to the fifth all over 10 minus five. So 10 miles five is just going to be five. And so then we could cancel these over again, just like we did before. Start to give us 200 for each. So now let&#x27;s see what we get are actually first, let&#x27;s factor out that 200 sweet have 200 we could also factor out a 1.5 raised to the Biff and doing that. Would he have 1.5 raised the fifth minus one already. So just writing get like this would be a little bit easier to plug into a calculator, I think so. We have 1.5 raise to the fifth. So about 1.27 by this one. That gives us about 10.27 that we multiply that by 1.5 race to the fifth, just about 0.35 and then multiply that by 200. So this is going to be somewhere in the ballpark off about 70 dollars and 52 cents. So that&#x27;s how much interest we would have burned on the 2nd 5 years now to estimate this rate of or the instantaneous story of change at T 05 so we could use the instantaneous story of change equation and then just keep taking smaller and smaller steps away from each other were we can average these two rates of change over here, So this is going to be approximately equal to to the average rate of change of 0 to 5, plus the average rate of change from 5 to 10 all over, too, because we&#x27;re just averaging those two. And this might be a good one to use cause notice that this one ends at five in this one starts at five. So the slope going from 0 to 5 and we have the soap coin from 5 to 10. So you would expect it to be maybe somewhere in the middle of those, so you just go ahead and plug these values. And so this would be approximately 55.26 plus 70.52 all over, too. And so let&#x27;s see what we get when we add these up. And so after we divide and add, that should give us something around $62.89. So this is what we would expect to be the instantaneous rate of change after five years, or how much money we would expect to earn after that fifth year.

Wen Zheng · Answer

here we have a compound interest problem and we&#x27;re going to do part a mostly with the non continuous compounding formula. And so we have, in an amount invested $3000 an interest rate of 5%. So we use the decimal point verify and a time of five years. So to find the final amount, if interest is calm, pounded annually, which would be an equals 11 time per year, we have a equals 3000 times one plus 0.0 5/1 raised to the power one times five. And we put that in a calculator and we get rounded to the nearest penny. $3828.84. Now we move on to the second part of part A, and we&#x27;re going to calculate it for semi annually, which would be two times a year so and equals two. So a equals 3000 times one plus 0.0 5/2 raised to the two times five, which would be 10 10 compound ings in the five year period. And we put that in a calculator and rounding to the nearest penny we get $3840.25 for the third part of part. They were going to calculate for monthly compounding, so that would be 12 times of compounding per year. So we have a equals 3000 times one plus 0.0 5/12 raised to the power 12 times five, which would be 60 compound ing&#x27;s and rounded to the nearest penny. We get $3850 and 20 and ah, seven cents seven cents. So we could see the amounts are going up a little bit each time as the number of compound ings increases for the fourth part were compounding weekly. There are 52 weeks in a year, so an equals 52. So we have a gal&#x27;s 3000 times one plus 0.0 5/52 raised to the power 52 times five, and that&#x27;s going to give us $3851.61. And for the fifth part of part, a daily compounding will use n equals 365. So we have a equals 3000 times one plus 0.0 5/3 65 raised to the power 3 65 times five. Put that into a calculator and we get $3852.1. And for part six, we&#x27;re going to switch to continuously compounded interest. So we have a different model for this. We have a equals a not times e to the R. T. So that&#x27;s going to be 3000 times E to the 0.5 times five, and we end up with $3852.8 not a lot more than the previous one. But it&#x27;s the highest you can get for continuously compounded interest compared to any of the other calm poundings now for Part B were to find the differential equation that fits the situation. So because we have this basic exponential growth type model here, it fits. The differential equation rate of change of A is proportional to a so rate of change of a equals K times A. Now we know the value of K will be our rate. And in our equation we had a equals 3000 times E to the 0.5 t, so the rate was your a 0.0 fight. So that&#x27;s R K. So now we have de a d t equals 0.5 times a, and we can go ahead and substitute our equation. And for a and we have 0.5 times 3000 each of the 0.5 t. All right, let&#x27;s multiply the 0.5 and the 3000 together, and we get 1 50 e to the 0.5 t. So that&#x27;s our differential equation for this situation. And for the initial condition. That would be what&#x27;s the amount when the time is zero and we know the amount invested was $3000. So at time zero the amount is 3000. That&#x27;s the initial condition.

Problem

(a) How long will it lake an investment to double…

Problem 21 Hard Difficulty

Answer

a)
(i) annual: $A(5)=\$ 3828.84$
(ii) semiannually =$\$ 3840.25$
(iii) monthly: $A(5)=\$ 3850.07$
(iv) weekly: $A(5)=\$ 3851.61$
(v) daily: $A(5)=\$ 3852.01$
(vi) continuously: $A(5)=\$ 3852.08$
b) $\frac{d A}{d t}=0.05 A(t)=150 c^{0.05 t} \quad$ initial condition: $A(0)=3000$

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Catherine R.

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a)(i) annual: $A(5)=\$ 3828.84$(ii) semiannually =$\$ 3840.25$(iii) monthly: $A(5)=\$ 3850.07$(iv) weekly: $A(5)=\$ 3851.61$(v) daily: $A(5)=\$ 3852.01$(vi) continuously: $A(5)=\$ 3852.08$b) $\frac{d A}{d t}=0.05 A(t)=150 c^{0.05 t} \quad$ initial condition: $A(0)=3000$

Calculus

Grace H.

Catherine R.

Kristen K.

Samuel H.

Precalculus Review - Intro

Functions on the Real Line - Overview

James StewartCalculus

Grace H.

Catherine R.

Kristen K.

Samuel H.

a)
(i) annual: $A(5)=\$ 3828.84$
(ii) semiannually =$\$ 3840.25$
(iii) monthly: $A(5)=\$ 3850.07$
(iv) weekly: $A(5)=\$ 3851.61$
(v) daily: $A(5)=\$ 3852.01$
(vi) continuously: $A(5)=\$ 3852.08$
b) $\frac{d A}{d t}=0.05 A(t)=150 c^{0.05 t} \quad$ initial condition: $A(0)=3000$

James Stewart

Calculus