Continuous compounding For the case in the following table,...

Continuous compounding For the case in the following table, find the future value at the end of the deposit period, assuming that interest is compounded continuously at the given nominal annual rate. (Click on the icon here in order to copy the contents of the data table below into a spreadsheet.) Amount of inititial deposit $800 The future value at the end of the deposit period is $ Nominal annual rate, r 15% Deposit period (years), n 6 (Round to the nearest cent.)

Transcript

00:02 In this question, we're given a couple of different investment scenarios, and we have to figure out a function for the interest.

00:11 We have to figure out how much we're going to have in the account after 5, 10, 30 years, and then we have to figure out how much it's going to take to double our investment.

00:19 So to start off, we have a principal of $500 with an interest rate of 0 .0075, compounded monthly, means an end value of 12.

00:29 Using the compound interest formula, we can get a function for this by substituting in our values.

00:35 That's going to be a of t is equal to 500 times the quantity of 1 plus 0 .0075 divided by 12, raise to the power of n times 12.

00:57 Or if i rewrite that, i will get a of t is equal to 500 times the quantity of 1.

01:05 1 .000 625 raised to the power of 12 times t.

01:16 I miswrote this here.

01:22 Now i'm going to use this function to find out how much money i'm going to have after 5, 10, and 30 years.

01:30 So if i substitute in 5 to that equation, i'm going to end up with $519 .10 after 5 years.

01:40 After 10 years, i will have 5 $538 .93.

01:47 And after 30 years, i will have $626 .12.

01:53 In order to double the money, i have to rearrange the compound interest formula.

02:00 To double the money, my desired amount over the principal is going to be two.

02:06 So i'm going to end up with the natural log of two divided by 12 times the natural log of 1 plus.

02:17 0 .0075 divided by 12.

02:21 And if i evaluate this, i'm going to find that it will take me approximately 92 years to double my money under these conditions, rounded to the nearest year.

02:36 In the second scenario, we have initial investment of $500, an interest rate of 0 .0075, but this time it's compounded continuously.

02:48 So we're going to use the compound interest formula, p, e to the rt, in order to get our function.

02:56 If i substitute in my values, i'm going to get that the function is a of t is equal to 500 multiplied by e, raised to the power of 0 .0075 times t.

03:12 Now using this function, i can evaluate how much money i'm going to have after 5, 10, and 30 years...

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