interest-if-1000-is-invested-in-an-account-that-pays-5-compounded-continuously-the-total-amount-at-i

Question

Interest If $\$ 1000$ is invested in an account that pays 5$\%$ compounded continuously, the total amount, $A(t),$ in the account after $t$ years is
$$A(t)=1000 e^{0.05}$$
a. Find the average rate of change per year of the total amount in the account for the first five years of the investment (from $t=0$ to $t=5$ ).
b. Find the average rate of change per year of the total amount in the account for the second five years of the investment (from $t=5$ to $t=10 )$ .
c. Estimate the instantaneous rate of change for $t=5$ .

Bobby Barnes · Accepted Answer

We&#x27;re told that if we invested $1000 into account, that pays 5% interest that compounds continuously with T being how many years we leave it in. Then we can see how much money we have in the account. By the equation eight TZ until 1000 times ive rates than 0.5 And we want to find the average rate of change off, how much money we have in the account in the 1st 5 years and the 2nd 5 years. And then they want to estimate the instantaneous rate of change for when t is equal to five. So let&#x27;s go ahead and plug everything into this equation we have up here. So we&#x27;re going to look at So instead of f, we&#x27;re going to use a so it&#x27;s gonna be a five minus a of zero all over five minus zero. Now, if we were to find a five. So first, let&#x27;s just pluck five and still be 1000 times e raised to the 0.5 times five. So that should be 0.25 and then minus 1000 raised to zero, which would just be one. And then in the denominator, we would have five. So first knows we could divide this five into what we have in the numerous s just gonna be too 100 for each of those. So we&#x27;re gonna have e raised the 0.25 times 200 then minus 200. So that&#x27;s going to be something around 50 six point 81 dollars Now for B, it&#x27;s going to be the same set up. We&#x27;re just going to use five intent. This times will be a of 10 minus a five all over 10 minus five. So plugging in 10 we would have 1000 e to the 0.5 minus 1000 e toothy 0.25 and then 10 minus five is just gonna be five. So again, we can go ahead. Divide those fives and each so we&#x27;d have 200. Reach those. And once we simplify this town, let&#x27;s see what we could get. So, e rates. Is there a point? Uh, minus. He raise to the 0.25 and we multiply that by 200 then divide not just multiplied by 200. So that&#x27;s going to be something around 72 point $94 so century will be just found was the average rate of change for each year. So I should probably do per year. So in the 1st 5 years we expected to increase by about $56.81. And in the 2nd 5 years, we expect each year for it to increase by about $72.94. Now for this next part, it says estimate the instantaneous tray of change at five. So they want to find exactly what would be our rate of change for how much money were earning after five years. Well, we could just plug everything into the instantaneous rate change equation and then just keep taking smaller, smaller values of H. But notice how in these last two questions, we found the average rate of change and it was centered around five. So one way we can estimate this is by just adding those two values together and then dividing by two. Because if we found the slope of the rate of change before that and estimated it and then we do it the same thing on the right, we would expect it to be something in the middle of those two values, so begin just right. This has to the average rate of change from 0 to 5, plus the average rate of change from 5 to 10 all over to so spoke everything good. It would be 56.81 plus 72.94 all over, too. And once we add these up, that should give. So that&#x27;s 129.75 and divide that by two. So after we&#x27;ve around, there should be something around 64 point 88 dollars per year. So after five years, we would expect us to gain about $64.88 for the remainder of that year.

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.

Regina Hays · Answer

okay. If we know, um, that we have $10,000 um, are invested, um, in an account that pays 2% continuously, um, compound is continuously, um um And, um, the total after after tea here&#x27;s is given by this function, ACE of T is equal to 10,000 e. Reyes. Who? That 0.2 times t. And the first thing we want to do is to find the average rate of change per year of the total amount, Um, in account for the 1st 10 years. Yeah, from T equal to zero to t equal to tend to the 1st 10 years. Okay. And so we know the average rate of change is given by our function evaluated at 10. Minus our function. Evaluated it zero all over, 10 minus zero. So we get, um, 10,000 e raised the 0.2 times 10 minus, um, each of the zero, which is, um, one. And this is gonna be divided by 10. So we get, um, $221.40 per year. So that&#x27;s how much is going thio the average or to change per year over the of the total amount in the account, so it&#x27;s gonna change $221.40 per year on average. Okay. And so then the next thing we wanted to you is we want to find the average rate of change. Um, per year, um, from T equal to 10 to t equal to 20. Okay. And so, once again, that&#x27;s the function evaluated 20 minus the function evaluated 10 all over 20 minus 10. So we&#x27;re gonna have 10,000 e raised to the 0.2 times 20 minus e raised to the 0.2 times 10 and that is gonna be divided by 10. And so when we put this in our calculator, we get $270.42 per year. So in the first 10 years, we on average, do a $221.40 per year. But in the later 10 years, we get an average rate of change of $270.42 per year. And then what we want to dio is too. Then find the instantaneous rate of change and at T equal to 10. And so that is going to be given by on. What we&#x27;re gonna do is we&#x27;re in use of formula and we&#x27;re gonna let h b 0.1 So we&#x27;re gonna evaluate our function at 10 plus 100.0 01 minus our function evaluated at 10 all divided by that 0.1 And you could d&#x27;oh! 0.1 All the fact is is you just need to make a choice. Smallest possible. And so we get, um, 10,000 times E to the 0.2 times 10 point, um 0001 minus e raised to the 0.2 times 10. And then this is all divided by 0.1 And we should get $244.28 per year at the 10th year.

Helen Telila · Answer

So for this problem, we given the equation for the amount of money that would be left in the account after a certain amount of time in time is measured and years. So don&#x27;t worry about the details about the and interest rate and stuff, because we&#x27;re really not gonna use that. It&#x27;s already included in the equation. So let&#x27;s just focus on the definition of the average change. A change of great weight and the general definition for that has changed. And a which is changing the amount of money in the account over change in time. So it&#x27;s a basic definition. So now, because we&#x27;re in part a were asked if eyeing the average change of great when time goes from zero Teo, it&#x27;s time go through ten years. Okay, I&#x27;m not writing the units here, but you can. So now I&#x27;m just going to substitute okay, Before I substitute, let me just make it clear what we&#x27;re doing. We&#x27;re doing a time team equals ten minus a. But time, people zero divided by change in time, which is ten years minus zero years. So now if you plug in the numbers what your heart is ten thousand Marius. It gives a ten thousand this Coleman for bows on just going toe. Put it outside of the bracket one point O too. Raise it to ten minus one point. Oh, shoo. Home time, Ciro, Reach to here. All right. Divided by ten. Okay, so if you run the numbers for this on your calculator, you point the way to a hey. No. So that&#x27;s, uh, the average rate of change for a year. Okay, so that&#x27;s for a now this trip part, where is the same? The only change is the time. Time goes from C equals head, too. T equals twenty. Okay, So what we&#x27;re gonna do is changing, eh? Just want money divided by change in time. This will be a twenty minus ten. What if I find my name? Okay, so we&#x27;re doing the same thing that we did earlier. For eight. Sometimes one going all to wrist you twenty minus one point two. Yes, he can divided by. So for this one, when you&#x27;re forgiven members, you should call for later. I own to be two hundred sixty six, knowing all to me. That&#x27;s the average rate of change going from T ten to twenty years service. This is for years. Remember, the unit here is near. So that&#x27;s why they keep writing for a year. And, um so this is dollars for a year. You want to put a dollar sign or you can put a dollar sign here? Not there. You&#x27;re so home. Note that in part A the time difference was ten and part B to the time difference is ten. But the change in the amount of money is actually slightly different rate. So here it&#x27;s two hundred, about two hundred eighteen. And, um sorry. I don&#x27;t know where that my and import be. It&#x27;s about two hundred sixty six, so that&#x27;s significant. But remember, this is what people call usually compounded interest, the power of compounded interest. But we&#x27;re not getting into that. But okay, let&#x27;s Newport Sea Sports. He is asking us to kwai the instantaneous weight of change, okay? Or to say instead, rate of change in on time. But this one is stay wine. A T equals ten. Yeah, see? So remember that the definition for instantaneous rate of change iss using limit right now. So we&#x27;re going to say this is limit of and C h my news here too. Fighting for our lives. The difference here. Age an age. We&#x27;re doing them. Limit us. A school&#x27;s here. So now again, we&#x27;re doing the same thing. We&#x27;re just plugging him the numbers. So a different color in housing. So the limit is out here. I&#x27;m not waiting that they met right now. It&#x27;s just focus on the equation. You have one week all too. Two plus age minus form old too. Team provided for H, right? Okay. But we know t equal stand so And we can take out team one point this fort in this both sides have the same numbers so we can pull it out to the side. Right. So just we write that nice you&#x27;re a hero to to the T and t is ten, right? And then what? You&#x27;re like, where&#x27;s is one point two bridge to the age miners. One right, provided by beach. Okay, so this is a constant rate. The only thing that&#x27;s changing as h So now let&#x27;s make a table for a small increments of age. One morning out of space years always threw it on the next page, so it&#x27;s safe for H. But start was large. Wally&#x27;s of H Great. Okay, so, comport. See us. This is the equation. So age over here, it&#x27;s too at H equals one, for instance, h People&#x27;s zero point one at age people&#x27;s zero point zero wine age people&#x27;s zero point zero zero one. You can keep going and going so and now the follies from equation that we found here. If you just plug in the volume H for different while he&#x27;s of age where this, for instance, for each people&#x27;s one bomb. Two for three point seven, eh? Or this one two for one for you fix three. Two no children, Hugh for six. And then I absolutely didn&#x27;t do thiss one. What a dead wass. I just like, made this one really training a murder just to see what happens, right? So I put an bites sirus one, too. Or why e got a two for one for me, really there. So it&#x27;s safe to say that of the average, not the average, that instantaneous change AT T equals ten. It&#x27;s about two hundred forty one Maureen, four dollars for a year, so that is at all this water

Amrita Bhasin · Answer

this question asked us to give unit and interpreter answer in terms of the balance in the given account. What we know is that the balance, the re average rate of change of the balance is Delta bi over Delta T. Obviously, Delta means change. This look of went to 1000 times 1.8 to the power of five minus 1/5 minutes here, which is obviously just divide by five. This is Oakland went to $93.87 per years. Four units are dollars per year. Now what we know is that this balance on the account increases by an average of this amount in dollars per year.

Amy Jiang · Answer

that&#x27;s what we&#x27;re gonna do and you go to a not eating a joke. 104 to 5 30 And we know a knot is equal to one Belden and you have, ah, compound interest of 4.25%. Okay, so that&#x27;s equal to distant over here and were given from T is equal to one in two years with chip. So we have a one is equal to 1000 you need to know 0.425 times one that&#x27;s equal to 1043.4 two. Okay, 2 to 1000. You did it open to a 4 to 5 times two. That gives me women&#x27;s Oh, it&#x27;s 8.72 Now we&#x27;re asked to find the average rate of change. That&#x27;s this. A 210 8.72 minus 10 for 3.42 over one, which is itch threats this equal to 45.3 hours for you

Problem

Sales The graph shows annual sales (in thousands …

Problem 29 Medium Difficulty

Answer

a) 56.8
b) 72.94
c) 64.88

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Margaret L. Lial, Raymond N. Greenwell, Nathan P. Ritchey

Finite Mathematics and Calculus with Applications

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Catherine R.

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a) 56.8b) 72.94c) 64.88

Finite Mathematics and Calculus with Applications

Grace H.

Catherine R.

Michael J.

Joseph L.

Precalculus Review - Intro

Functions on the Real Line - Overview

Margaret L. Lial, Raymond N. Greenwell, Nathan P. RitcheyFinite Mathematics and Calculus with Applications

Grace H.

Catherine R.

Michael J.

Joseph L.

a) 56.8
b) 72.94
c) 64.88

Margaret L. Lial, Raymond N. Greenwell, Nathan P. Ritchey

Finite Mathematics and Calculus with Applications