Previous ProblemProblem ListNext Problempoint) Exercise 4)An investor makes three deposits into : fund Ine end 01 2, nomina rate of disccunt converible quarterl} AN...

Solve each problem. See Example $9 .$ Compounding contimuously. If $\$ 500$ is deposited in an account paying $7 \%$ compounded continuously, then how much will be in the account after 3 years?

We once again working with our compound interest formula. So start with formula. Humanity account is equal to the principal times one plus R Divided by end, which is our rate divided by a number of compounds raised to the number of compounds times time then we're gonna substitute into that formula are no values. We know that our principle is the initial amount of the amount we deposit, which is 39,500 in this case. And our interest rate is that 4.9%. Careful with that. Don't miss that 09 and it's being compounded daily says 365 times a year can no, of course, reason that exponents there's or 3 65 again, we go back to oppression. We find a number of years, which is nine. And now we're ready to simplify by bloodiness into our calculator. Now I get a grand total of $57,075 and 79 cents

So if 7000 is the positives. So if you have a principle of 7000 in an account paying 8% so our rate is 8% compound and continuously, then what is the amount? After four years, your cheesy could afford now for our compound continuously, we have a physical to P E R T, so it's plugged in. What we know. We have 7000 kinds e to the 0.8 times four. Plug this into my calculator. Let's see what we get 0.8 times for gives me 9639.89

Okay in this situation, Kurt wants to buy a truck um and the truck is going to cost $30,000. So That's the amount of the truck. Um he currently has $27,000. Um and he's going to invest it at 4% quarterly and we want to find out how many years it is because it's giving us quarterly and not continuously we're going to use our formula A. Is equal to p times one plus the are over the end to the uh N. T. All right. So that's our formula. So we have all these parts except for for the T. That's what we're solving for. We also before we do this we want to change our rate into a percent 4% as a decimal is 40.4. Okay, that's going to help us. All right. R. A. Is 30,000. Our principal RP is 27,000. That's how much kurt has. All right. One plus r over end. So one plus the rate as a decimal is .04. And quarterly is is the amount is four times a year. So the N. Is for alright, the end again for the exponents for we are solving for tea. Okay, so this looks intimidating because you're solving for an exponent. But what we want to do is isolate the portion uh with the exponents And then use our law algorithm properties to solve it. Okay, so the first step I'm going to do is divide both sides by the 27,000. All right, well that way we can get our parentheses by itself. All right. Um and you can do this in a calculator. You can get a decimal. It comes out to 1.1 repeating. I'm just going around it to 1.11 for now. All right. Um you can do a little bit of work inside your parentheses. 2.04 divided by four is point a one and one plus point no one would just be 1.1. Okay, um to the fourty. All right. And we learned the way that we can knock down exponents is by logging both sides. Okay, so our next step is to do a arl algorithms and what you do to one side, you're going to do the other side. So I get log Of 1.1, 1 is equal to the X moment comes out in front of the log so four t Log of 1.01. Yeah. Okay. Now I'm solving for TI want to get it by itself so I need to divide away this log 1.1. All right, what do I do to one side I do to the other side, log 1.01. And thats calculator work. All right, so get your calculator do log 1.11 divided by log 1.1. And it comes out to be I'm going to rounded to three decimal places for now I know the questions as round to the nearest 10th but I don't want to be um I don't want to round my intermediate steps too much. That comes out to 10.488 is equal to four T. Right now I just have one step equation I need to solve for T. So I have to divide both sides by four. In 10.44, sorry, 488, divided by four is going to come out to be approximately 2.6. We're rounding to the nearest 10th. Alright, it came out to be 2.6-2, so I'm gonna leave it as 2.6 years. All right.

The question as to solve are given the equation a equals p Times EADS is a power of RT than to find the rate of which $1000 deposited with double in three years. So given that a equals p times e r T, we can isolate our by first dividing a by P. Then we can take the natural log of both sides, and we take the natural log because the natural log off e to any power or to any exponents equals that exports. So now the equation becomes Ellen off a over p equals R T. Then we divide both sides by t and we get a overpay divided by T equals R. Now that we have the equation, we want to find the rate at which $1000 deposited would would double in three years. So that means at the end will have $2000 because 1000 times two is $2000 in this equation, T ever. But it represents the amount of years or three years A is our future value, or $2000 P is a principal value or deposited value, which is $1000 so plugging in the values we have Ln of 2000 divided by 1000 all over three years, or Ln of two divided by three, and we would get 0.231 or 23.1 percent.