Problem 42 - College Algebra with Corequisite Support

Question

Suppose an investment account is opened with an initial deposit of \(\$ 12,000\) earning \(7.2 \%\) interest compounded continuously. How much will the account be worth after 30 years?

Step-by-Step Solution

Answer

The account will be worth approximately \( \$104,092.80 \) after 30 years.

Step 1: Identify the Formula

In problems involving continuously compounded interest, use the formula \( A = Pe^{rt} \), where \( A \) is the amount in the account after time \( t \), \( P \) is the principal amount (initial deposit), \( r \) is the annual interest rate, and \( t \) is the time in years.

Step 2: Insert Known Values

Given \( P = 12,000 \), \( r = 0.072 \) (convert \(7.2\%\) to decimal), and \( t = 30 \), insert these into the formula: \( A = 12,000e^{0.072 \times 30} \).

Step 3: Calculate the Exponent

Calculate the exponent: \( 0.072 \times 30 = 2.16 \). Then, calculate \( e^{2.16} \). You can use a calculator for this step.

Step 4: Compute the Final Amount

Using a calculator, find \( e^{2.16} \approx 8.6744 \). Substitute this back into the formula: \( A = 12,000 \times 8.6744 \).

Step 5: Complete the Calculation

Calculate \( 12,000 \times 8.6744 = 104,092.80 \). Thus, the amount in the account after 30 years is approximately \( \$104,092.80 \).

Key Concepts

Understanding Exponential Growth

Continuous compounding of interest illustrates the concept of exponential growth. This occurs when the growth rate of a value is proportional to its current size. In the context of an investment, this means that the more money you have, the faster it grows over time.

This idea is crucial because money grows not just on the initial principal, but also on the accumulated interest over previous periods. This compounding effect leads to exponential growth, which can be visualized using the formula for continuously compounded interest:
\[ A = Pe^{rt} \]- **Base "e"**: The mathematical constant "e" serves as the base rate of growth shared by all continually growing processes. This constant is approximately 2.71828.- **Time as an exponent**: In continuous compounding, time is an exponent in the formula, highlighting its pivotal role in exponential growth. The more time that passes, the greater the amount you will accrue.By understanding exponential growth, you can appreciate why investments grow at such a rapid rate when continuously compounded, providing a clearer picture of financial potential over time.

Interest Rate Calculation Made Easy

Calculating the interest rate is a fundamental part of understanding how your investment will appreciate over time. Continuously compounded interest calculations require you to express the interest rate as a decimal. For example, a rate of 7.2% becomes 0.072 when converted.

### Key Steps to Calculate Interest Rate: - **Identify the percentage**: Understand what percentage of growth your investment involves, in this case, 7.2%. - **Convert it to a decimal**: Simply divide by 100 (7.2 ÷ 100 = 0.072) to simplify the calculations in any formula. This conversion helps you plug the interest rate directly into the equation for compounding interest calculations. By grasping this simple conversion, you eliminate common errors and will have a precise input to compute your future investment value.
Remember, accurate conversion ensures that your interest rate is perfectly poised to determine your investment's potential to grow, keeping it in line with exponential growth principles outlined earlier.

Delving into the Compound Interest Formula

The formula for continuously compounded interest, \( A = Pe^{rt} \), provides a powerful tool to calculate future investment value. Understanding each component of the formula will help you see how changes to initial deposit, interest rate, and time impact the outcome.
### Components of the Formula:

• **\(A\):** The future amount of money you will have in the account after interest.
• **\(P\):** The principal, or initial amount of money, which in this case is $12,000.
• **\(r\):** The annual interest rate as a decimal.
• **\(t\):** Time in years during which the money is invested.

Here’s how the formula works:- **Exponentiation and "e" factor**: Time and rate act as an exponent tied to "e", emphasizing the growth factor.- **Primary Calculation**: Multiply the principal by the result of this exponentiation.
This formula highlights the power of exponential growth in finance, showing how a small change in time or rate can significantly affect the end figure. Mastering this equation allows investors to forecast returns accurately, measure compounding effects over time, and make informed financial decisions.