a-bank-account-earns-5-annual-interest-compounded-continuously-money-is-deposited-in-a-continuous-ca

Question

A bank account earns $5 \%$ annual interest, compounded continuously. Money is deposited in a continuous cash flow at a rate of  $\$ 1200$ per year into the account.$
(a) Write a differential equation that describes the rate at which the balance $B=f(t)$ is changing.
(b) Solve the differential equation given an initial balance $B_{0}=0$.
(c) Find the balance after 5 years.

Cyrielle Lorio · Accepted 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.

Zach Wiebe · Answer

All right. Here we have Chapter nine, Section one. Problem number four, Um, And for this problem, we need to write a differential equation. We have a big account, and the money in it earns interest at a continuous annual rate of five times the current balance. Um, so we want to write a differential equation for the balance, which is B as a function of time, key in years. Um, so we&#x27;re looking for the rate change of be, which means we&#x27;re finding Do you be t t And we&#x27;re writing a differential equation for be DT um, so every every year, every unit of tea, um, we have 5% of the balance. He added. So, um, this DBT T is how much is added every year, and that&#x27;s 5% of the balance. So you can get that by just multiplying the balance by 5% or as a number, that would be 0.5 which is 5% of one. Um, and it&#x27;s that simple. There is transferred to the problem, Um, again, every year, this 5% 0.5 of the money in the account has added to the account and As the money in the account increases, so does the interest. See, there you go. 0.5 that is.

Cyrielle Lorio · Answer

Okay, So what we have here is a bank account that is compounded continuously at an interest rate off 10% or a zero point then and then initial violence in the room. So money is also deposited to the pump account at a continuous rate of $1000 per year. So we are required to get be as a function of f of the Intersil for its solution. So now we can represent being be over BP as point. Then be this 1000. So we want to transfer this so the left side. And we do this by multiplying both sides by one over point. Then be less 1000. Now we&#x27;ll have baby over point then B plus 1000 bebe physic was one. Now we want to multiply both sides. Baby, be. We&#x27;ll get Bebe over point, then be this 1000 Is it? Well, so bp so integrating Both sides will have one over 0.1 and and off point. Then be this 1000. Is it? Well, could be less See one. So we multiply both sides by point then and we&#x27;ll have and then a point. Then be plus 1000 is equal to a point then the plus 0.1 C one. So remember that this is just a constant. So we can therefore 2% this us just see so making both sides of power off e will have books are you You will have a point then be plus 1000 is equal though theories toe point then be let&#x27;s see So again see is just ah constant value so it can pick up any by use and we can represent our equation. This point then be plus 1000 is equal to C e Raise the point and be if a further isolate b will have B is a well too one over, then one over point then see theories 2.1 B minus. Then thou sent again. This can&#x27;t just be represented a sea. So our final solution is being is equal to C series the 0.1 b minus then thou son, So we want to find the value of C and we do this by making use off the initial conditions give in from the problem. So at T is equal to zero. B is equal to zero. So substituting this hour, we should will have. Zero is equal to C E. Raise 2.1. Thank zero minus than out, son. So this will just be equal Toe one and solving will have seen is equal to 10,000. So did I think Alverson, who should have be off being is equal to then thou son. Here is the 0.1 B minus, then absent. So this will be our final answer in our part together to new one.

Cyrielle Lorio · Answer

Okay, So what we have here is a bank account that goes continuously at an interest rate of 5% or 0.5 and supposed to make an initial deposit a year 2010 with an amount of $2000. So with this problem that we will consider when did then that&#x27;s the starting year. So this will be our B 00 So we are required to get em off B. And for another view, we are required to get the solutions at em off. 30 given the industry 0.5 and zero point then so for hours. And you should Okay, so we can express B m off over the is equal to 0.5 m. Now we multiply both sides by one over m and our insulting equations. Bm over M V P is equal to 0.5 Now we multiply both sides by BP be doing that. We&#x27;ll have the m over em is equal to 0.5 bp. So integrating both sides of the equation, our resulting equation will be and in a M is equal to 0.5 b. Plessy, then raising boots said to the equation, Toe e will have em is equal to erase 0.5 b. Let&#x27;s see. So remember that. See who is just a post on so it can take anybody. You therefore we can represent their equation. Us I m is a well, Cassie areas to 0.5 feet. So this is now the general. So you Sean Now we want to find this particulars to new showed by finding the value of C. So we make yourself the initial conditions. Given a year 2010 which is at B is equal to zero C m. I mean is it will do without sun. So we substitute this over general solution and we&#x27;ll have 2000 is equal to see a raise to 0.5 10 0 So this will just be equal. The one therefore CE sequel do without sun. We cannot rewrite our solution as and is equal to Oh, sorry em off B, is it? Well do 2000 e raise 0.5 feet. Okay, So for a letter B, we want to find its Nushawn at your 24 B, which is 30 years after 20 then a at a rate off 0.5 So we do this by substituting the values to our particulars who shone so em off 30 is equal to 2000 e. J. C. 0.5 times 30 which is approximately equal. So 8000 963 point there we eat for the next one we were asked to get its solution at P is equal to 30 but an interest rate off 10% or zero point then. So our aggression, therefore would be em off. B is equal toe 2000 e raise 2.1 be in substituting 30 Could be will have em off 30 is equal toe 2000 Erase a 0.1 times 30 which is approximately equal to 40,000 171.7 This is our answer

Heather Zimmers · 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

Money in an account earns interest at a continuou…

Problem 10 Medium Difficulty

Answer

a. $1200+0.05 B(t)$
b. $-24000+24000 e^{0.05 t}$
c. $-24000+24000 e^{0.25} \approx \$ 6816.61$

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Deborah Hughes-Hallett, Patti Frazer Lock, Andrew M. Gleason

Applied Calculus

Catherine R.

Kayleah T.

Caleb E.

Samuel H.

Integration Review - Intro

Definite vs Indefinite

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a. $1200+0.05 B(t)$b. $-24000+24000 e^{0.05 t}$c. $-24000+24000 e^{0.25} \approx \$ 6816.61$

Applied Calculus

Catherine R.

Kayleah T.

Caleb E.

Samuel H.

Integration Review - Intro

Definite vs Indefinite

Deborah Hughes-Hallett, Patti Frazer Lock, Andrew M. GleasonApplied Calculus

Catherine R.

Kayleah T.

Caleb E.

Samuel H.

a. $1200+0.05 B(t)$
b. $-24000+24000 e^{0.05 t}$
c. $-24000+24000 e^{0.25} \approx \$ 6816.61$

Deborah Hughes-Hallett, Patti Frazer Lock, Andrew M. Gleason

Applied Calculus