Problem 6 - Applied Calculus

Question

Suppose $$\$ 1000$$ is invested in an account paying interest at a rate of \(5.5 \%\) per year. How much is in the account after 8 years if the interest is compounded (a) Annually? (b) Continuously?

Step-by-Step Solution

Answer

(a) \$1518.10; (b) \$1553.60.

Step 1: Understand the Formula for Annually Compounded Interest

When interest is compounded annually, the formula to calculate the total amount is given by: \[ A = P(1 + r)^n \] where \( A \) is the amount of money accumulated after \( n \) years, including interest, \( P \) is the principal amount (initial investment), \( r \) is the annual interest rate (decimal), and \( n \) is the number of years.

Step 2: Calculate the Annual Compound Interest

Given \( P = 1000 \), \( r = 0.055 \), and \( n = 8 \), substitute these values into the formula: \[ A = 1000(1 + 0.055)^8 \] From this, calculate: \[ A = 1000(1.055)^8 \] \[ A \approx 1000 imes 1.5181 \] \[ A \approx 1518.10 \] Therefore, there will be approximately \( \$ 1518.10 \) in the account after 8 years.

Step 3: Understand the Formula for Continuously Compounded Interest

When interest is compounded continuously, the formula to calculate the total amount is given by: \[ A = Pe^{rt} \] where \( e \) is the base of the natural logarithm (approximately 2.71828), \( A \) is the amount of money accumulated after \( t \) years, \( P \) is the principal amount, \( r \) is the annual interest rate, and \( t \) is the number of years.

Step 4: Calculate the Continuous Compound Interest

Given \( P = 1000 \), \( r = 0.055 \), and \( t = 8 \), substitute these values into the formula: \[ A = 1000 \times e^{0.055 \times 8} \] Calculate the exponent: \[ 0.055 \times 8 = 0.44 \] Therefore, \[ A = 1000 \times e^{0.44} \] Using a calculator to find \( e^{0.44} \): \[ e^{0.44} \approx 1.5536 \] Thus, \[ A \approx 1000 \times 1.5536 \] \[ A \approx 1553.60 \] Therefore, there will be approximately \( \$ 1553.60 \) in the account after 8 years if compounded continuously.

Key Concepts

Understanding Continuously Compounded Interest

Continuously compounded interest is a concept where interest is calculated and added to the principal balance at every possible instant. This method uses the formula \( A = Pe^{rt} \), where:

• \( A \) is the amount of money accumulated after time \( t \),
• \( P \) is the principal amount,
• \( r \) is the annual interest rate expressed in decimal form,
• \( t \) is the time the money is invested for, in years.

The key element that makes continuous compounding unique is the use of the mathematical constant \( e \), which is approximately equal to 2.71828.

In scenarios where banks or investors want to compound interest as frequently as possible, they use this continuous method, making use of modern computational capabilities.

For example, if you were to invest $1000 at an annual interest rate of 5.5% for 8 years, using continuous compounding, you can calculate \( A = 1000 \times e^{0.055 \times 8} \). This results in roughly \( 1553.60 \), showing the power of continuously letting your investment grow.

Annually Compounded Interest Explained

When interest is compounded annually, it means that the interest is added to the principal once a year. This conventional method is easier to calculate and understand. The formula used here is \( A = P(1 + r)^n \), where:

• \( A \) is the accumulated amount after \( n \) years,
• \( P \) is the initial principal invested,
• \( r \) is the annual interest rate expressed as a decimal,
• \( n \) is the number of years the money is invested for.

To better understand, consider investing $1000 at an annual interest rate of 5.5% for 8 years. By substituting these values, the calculation would look like: \( A = 1000(1.055)^8 \).

This exponentiation means you multiply the factor \( 1.055 \) by itself 8 times. Using this annually compounded method, the total accumulated amount would be approximately \( 1518.10 \).

This traditional method illustrates how compound interest grows over time, though less rapidly than continuous compounding.

Calculating Interest Rates Made Easy

Calculating interest rates effectively requires understanding the role they play in financial growth. To convert a percentage to a decimal, you need to divide by 100. So for an interest rate of 5.5%, the decimal form is \( 0.055 \).

Recognizing the decimal form is crucial when inserting rates into compounding formulas. For instance, in both continuously and annually compounded scenarios, the consistent use of the decimal form simplifies calculations.

Understanding the importance of accurate rate calculation will help you predict investment growth accurately. This skill ensures that you can compare different compounding techniques or evaluate potential return on various financial opportunities.