Find the required annual interest rate to the nearest tenth of a percent for 65,000 dollars to g

step 1 Hello. So we're going to be using the formula here. A is equal to p times the quantity 1 plus R

Find the required annual interest rate to the nearest tenth...

Key Concepts

Compound Interest

Compound interest involves the process of earning interest on both the original principal and the interest that has been accumulated from previous periods. This results in exponential growth over time as interest is compounded at regular intervals.

Periodic Interest Rate

When interest is compounded more frequently than once per year, such as monthly, the annual interest rate is divided by the number of compounding periods (in this case, 12) to obtain the periodic interest rate. This period rate is then used in the compound interest formula to account for each compounding period.

Solving for the Interest Rate

In situations where the initial amount, final amount, compounding frequency, and time period are provided, the compound interest formula can be rearranged to solve for the unknown interest rate. This involves isolating the rate term, applying the appropriate root to remove the exponent, and then converting the periodic rate back to an annual rate.

Transcript

0:00 Hello.

00:01 So we're going to be using the formula here.

00:03 A is equal to p times the quantity 1 plus r over n to the nt.

00:07 And here we're asked to find the rate r.

00:10 So we're given the amount we have after t years is $6 ,500 or $65 ,000, $6 ,7 .883 .91.

00:27 So there's our a.

00:28 And that's going to be equal to, while our principal p is 65 ,000, so it's equal to 65 ,000 times, well, one plus r over n.

00:41 We're here, n is going to be equal to 12.

00:43 So we have one plus r over 12, and then that is then raised to the, well, to the 12 times 0 .5 power.

00:57 So therefore, we divide.

01:00 Through by 65 ,000 and we get um well we get um six five seven eight three point nine one divided by sixty five thousand is going to be equal to well one plus r over twelve to the sixth power um so we can solve for our here by taking the natural law of both sides because doing so we get the natural log of the right side is going to be equal to the natural log of the right side to a power and you know if we have a log of something to a power the power can come out front as a coefficient so therefore we have the natural log of the left is equal to six times the natural log of the right then dividing by six we get that the natural log of 1 plus r over 12 is going to be equal to one -sixth times the natural log of, well, of the 65 ,783 .91 divided by 65 ,000.

02:24 I would just leave it just like that to put in our calculator.

02:27 But then converting to an exponent, right? we have, well, this would be log base e.

02:33 So therefore, natural log of 1 plus r over 12...

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