Calculating Continuous Compound Interest: A Step-by-Step Guide

Naomi is planning to invest $8,700 in an account that compounds interest continuously. She wants the investment to grow to $13,000 over a period of 10 years. To determine the required interest rate, we need to use the formula for continuous compounding:

Continuous Compounding Formula

The formula for continuous compounding is:

[ A Pe^{rt} ]

A is the amount of money accumulated after time t (the future value) P is the principal amount (the initial investment) r is the annual interest rate (as a decimal) t is the time the money is invested for in years e is the base of the natural logarithm, approximately equal to 2.71828

In this case, A $13,000, P $8,700, and t 10 years. We need to solve for r.

Solving for the Interest Rate

We start with the equation:

[ 13000 8700e^{10r} ]

First, divide both sides by 8700:

[ frac{13000}{8700} e^{10r} ]

Calculate the left side:

[ frac{13000}{8700} approx 1.49425 ]

Now we have:

[ 1.49425 e^{10r} ]

Next, take the natural logarithm of both sides:

[ ln(1.49425) 10r ]

Calculate (ln(1.49425)):

[ ln(1.49425) approx 0.400 ]

Now solve for r:

[ r frac{0.400}{10} approx 0.0400 ]

Express r as a percentage by multiplying by 100:

[ r approx 4.00 ]

Rounding to the nearest tenth of a percent, the required interest rate is:

4.0%

Alternative Methods

There are a few alternative methods to solve this problem, including using a financial calculator or spreadsheet functions. One of the approaches is to use logarithms directly to solve for the rate.

Using Logarithms

The problem can also be solved using logarithms. Given the equation:

[ 570 4401 left(frac{j}{4}right)^{60} ]

where j is the nominal annual interest rate, and we want to solve for j.

First, divide both sides by 4401:

[ frac{570}{4401} left(frac{j}{4}right)^{60} ]

Calculate the left side:

[ frac{570}{4401} approx 0.12949 ]

Now take the 60th root of both sides:

[ left(0.12949right)^{frac{1}{60}} frac{j}{4} ]

Calculate the right side:

[ left(0.12949right)^{frac{1}{60}} approx 1.00432 ]

Solve for j:

[ frac{j}{4} 1.00432 ]

[ j 4 cdot 1.00432 approx 4.01728 ]

Rounding to the nearest tenth, the interest rate is approximately:

1.7%

Homework and Learning

For homework problems like this, it's important to do the work yourself. Many instructors expect you to use tools like a financial calculator or spreadsheet functions, such as Excel’s RATE function. This not only helps you understand the concepts better but also prepares you for real-world applications.

Remember, while it might be tempting to let someone else do your homework, the true learning comes from working through the problems step-by-step. If you get stuck, don't hesitate to ask for help from your instructor or peers.

When you solve these problems, you'll gain a deeper understanding of exponential growth and continuous compounding, which are important concepts in finance and investment.