To solve each problem, refer to the formulas for compound inte...

Question

To solve each problem, refer to the formulas for compound interest. A = P (1 + r/n)^tn and A = Pe^rt Find t, to the nearest hundredth of a year, if $1786 becomes $2063 at 2.6%, with interest compounded monthly.

Accepted Answer

Hello everybody. I am doing today. So we're going to be looking at this math question that states calculate the value of T rounded to the nearest 100th of a year for which $8472 will grow to $11,024 at a monthly compounded interest rate of 7.3%. Now, the answer choices are a 3.62, 62 years, B 2.62 years, C 2.92 years and D 4.62 years. Now, when it comes to a question like this, talking about interest rates and over time, we have a formula that states A is equal to p multiplying a parentheses with one plus R divided by N inside of that parentheses raised to the exponent N multiplying T. Now what do these variables mean? Well, the A is the final amount in our case is basically what amount do we want to grow to? Our case is 11,024 P is the initial amount or the principal. So what, what we start with? So it was 8472. The R is the annual interest rate. And the end is the number of times that our interest is compounded per year. And finally, our team is our time in years. So let's plug all those values in now that we know what each variable stands for. So we'll have a formula of our A which was 11,000 24 equals P which is the initial amount. And for us, that's multiplying a parentheses with one plus R, which is the rate. In our case, they tell us our interest rate was 7.3% which when turned into a decimal would be 0.73. Since going from a percent to a decimal, we just moved the decimal point over to the left twice. So we'll have back in our formula one plus 0.73 divided by the end, which is 12 because they tell us that we are being compounded monthly. So there are 12 months in a year. So that is 12 times and that's raised to the exponent N T. So again, we'll have 12 and we'll multiply it by, by T we're solving for T. So we can't plug in anything yet. So now it's just a matter of uh simplifying and trying to solve for team. So the first thing I'll do is I'll divide by the 8472. So I'll have, I'm gonna move over and I'm gonna rewrite this and I'll have and our parentheses, one plus 0. divided by 12, raced to the exponent 12 T and that'll be equal to 1.3 oh 573. And that's uh the answer after dividing by the 8472. So now I'm going to take the natural log of both sides to help us with the exponents. So I'll have a or natural log of one plus 0.73, divided by 12 race to the exponent 12 T. And that'll be equal to the natural log of 1.3 oh 1227573. And now we can recall a rule that we have for algorithms on how to manipulate them and basically how, what, what we can do with an exponent when we have an, an exponent with an al uh algorithm. So in this case, let's um imagine you have the natural log of A to the C that can be rewritten as C multiplying the natural log of A. So all you do with the exponent is bring it over in front of the log and multiply it. So our exponent was 12 T. So we can bring that over in front and we'll have 12 T multiplying the natural log of one plus 0.73 divided by 12. And that'll still be equal to the natural log of 1.3 oh 1227573. And now we can divide by the, we can bring over the L N from the left-hand side to the right hand side by dividing. So we'll have 12 team on the left hand side. Now equals, and then we'll have an enumerator, other fraction on the right hand side, we'll have N of 1.3 oh 1227573. And that'll be divided by the natural log of one plus 0.73, divided by 12. And now I can bring the 12 over from the left hand side to the right hand side by once again, dividing by 12. So I'll have T is equal to. And in this case, I'm just going to multiply the right hand side by the fraction one divided by 12. So I have 1/12, the fraction 1/12, multiplying the natural log of 1.3 oh 1227573. And that natural log is being divided by the natural log of one plus zero, 00.73, divided by 12. And now it's a matter of just plugging all of this into our calculators. So we'll have to plug in our natural log in the numerator, our natural log in the denominator and then divide that answer by 12. And once we do that, we'll have that T is equal to 3. years. And that would be the time for this question, the answer which matches with answer choice. A so I hope that was helpful and until next time.

To solve each problem, refer to the formulas for compound interest. A = P (1 + r/n)^tn and A = Pe^rt Find t, to the nearest hundredth of a year, if \$1786 becomes \$2063 at 2.6%, with interest compounded monthly.

Key Concepts

Compound Interest Formula

Solving for Time in Exponential Equations

Interest Rate and Compounding Frequency

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