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Suppose that you invest S6000 in an account that offers & nominal rate of annual interest 2% compounded continuously. P(t) Pekt What will your investment be wor...

Question

Suppose that you invest S6000 in an account that offers & nominal rate of annual interest 2% compounded continuously. P(t) Pekt What will your investment be worth in 10 Years? Remember moneyP(t)P(10)6) How long would it take for your investment to double? Round your answer tO cwo decimal places_yearsWhat is the effective annual vield, EAY, ofan account earning 2% compounded continuously?EAY =

Suppose that you invest S6000 in an account that offers & nominal rate of annual interest 2% compounded continuously. P(t) Pekt What will your investment be worth in 10 Years? Remember money P(t) P(10) 6) How long would it take for your investment to double? Round your answer tO cwo decimal places_ years What is the effective annual vield, EAY, ofan account earning 2% compounded continuously? EAY =



Answers

The amount of money, $A(t),$ in a savings account that pays $6 \%$ interest, compounded quarterly for $t$ years, with an initial investment of $P$ dollars, is given by
$$A(t)=P\left(1+\frac{0.06}{4}\right)^{4 t}$$
If $\$ 500$ is invested at $6 \%$, compounded quarterly, how much will the investment be worth after 2 yr?

We're going to use our compound interest formula we're going to solve for t the number of years it takes when you start with $100 end with $450 compounding four times a year at a 6% interest rate. So let's go ahead and substitute all the numbers into that equation. 4 50 is the final amount. 100 is the starting amount, that principle. And we have one plus 0.0 6/4 raised to the power 40. We want to solve this for tea. What if we simplified one plus 10.6 divided by four just to make it easier to write, it will be 1.15 Sweet 450 equals 100 times 1.15 raised to the fourty. Okay, so this is an exponential equation, and the first step in solving it will be two to divide both sides by 100. So we get 4.5 equals 1.15 to the power 40. Now is the time to use logarithms so we can take the log of both sides. I'm going to use law based 10 but you could use natural log or any other base you like. So the log of the left equals the log of the right, and then we can use the power property of logarithms and bring the fourty out to the front. So now we have log of 4.5 equals 40 times the log of 1.15 Now, if we're trying to solve for T, what we need to do is divide both sides by four and divide both sides by log of 1.15 So that gives us T equals log of 4.5, divided by four log of one point 015 Now let's go ahead and put that in a calculator and approximate it and we can round it to the nearest well, let's maybe rounded to the nearest 10th so we have log of 4.5. I don't think the problem gives us directions about rounding. So, perhaps, and is that, But maybe we'll round to the nearest 10th. 25.3 is the number of years that would take

We have. The compound interest formula written here as a f T is equal to p times one plus are over end to the N t. And what all of these things represent A of T. Here is the value of your investment. After a certain period of time, he is the initial investment. Uh, are is Thebes rate of interest. Um, that is given to you on your investment and is how many times the interest is calm, pounded yearly and t is the amount of time in years that you have been collecting interest. So we want to know Here is we have an equation. 10,250 times one plus 0.0 for over 12 to the 120. And we want to know how much is the what is the value of this investment here? So since we're looking for the value, we're looking for a of T, which is pretty simple. All we would have to dio is just plug this into a calculator. D'oh! 10,250 ah times one plus 0.0 for over 12 to the 120. And when you put that into a calculator, you end up getting, uh, we're going to round to the nearest sent. You end up getting $15,000. 15,000 281 281 and three cents. So the value of this, uh, investment account here would be $15,000. 280. $15,281.3.

You have $53,500. You are receiving an interest free of 5.5%. You're receiving interest quarterly. And you want to know how much money you have in six and one half years. Okay, so 5.5%. That's 5.5%. And the decimal version then this point oh 55 Quarterly means you're going to receive interest four times a year And 6.5 years is the same as 65. So let's fill everything into your compound interest formula. And let's simplify this a little bit extras. Who were there. Okay, so let's check point 055 divided by four And add one to it. So we have 1.1375. Now it's important that you don't around or if you're going to be dipping yourself out of some money And then four times 6 five is 26. So we're going to 53,500 Times 1.1375. Raised to the 26th power. It's gonna give us 76305 point go at three decimal places because this is money. So we have to run to the nearest sent. So you will have $76,305 and it's not comma period 59 cents.

For this problem, we want to find the yearly interest rate Um that's earned by an investment that doubles in 10 years. So we have that is doubling in 10 years. So we have it's two times the initial investment may not and that's equal to a not E to the R. That's what we're trying to find. And it does that in 10 years. So because it's doubling, we go from me not to to A not, which means we just divide by a knot and reluctance to, then we can take the natural log of both sides. Natural log of E to anything is the anything we get rid of that base e. So now we have the natural log of two equals 10. Are so are is going to be the natural log of two divided by 10. But that tells us is that if we doubled our amount in 10 years, multiplying this time 200, we see that our interest rate must have been 6.9, or 6.9%, depending on how you around


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