Problem 493 - Precalculus

Question

Suppose an investment account is opened with an initial deposit of \(\$ 10,500\) eaming 6.25\(\%\) interest, compounded continuously. How much will the account be worth after 25 years?

Step-by-Step Solution

Answer

The account will be worth approximately $50,103.48 after 25 years.

Step 1: Understand Continuous Compounding Formula

The formula for continuous compounding is given by \( A = Pe^{rt} \) where \( P \) is the principal amount, \( r \) is the interest rate, \( t \) is the time in years, and \( e \) is the base of natural logarithms (approximately 2.71828). \( A \) is the amount after time \( t \).

Step 2: Identify Known Variables

From the problem, we know that \( P = 10,500 \), \( r = 0.0625 \) (converted from percentage to decimal), and \( t = 25 \) years.

Step 3: Substitute Variables into the Formula

Substitute the values into the formula: \( A = 10,500 \times e^{0.0625 \times 25} \).

Step 4: Calculate the Exponent

Calculate the product of the interest rate and time: \( 0.0625 \times 25 = 1.5625 \).

Step 5: Compute the Exponential Expression

Evaluate \( e^{1.5625} \). Using a calculator, \( e^{1.5625} \approx 4.77176 \).

Step 6: Compute the Final Amount

Now, multiply the principal by the exponential result: \( A = 10,500 \times 4.77176 \approx 50,103.48 \).

Step 7: Interpretation of Result

The account will be worth approximately \( \$50,103.48 \) after 25 years.

Key Concepts

Exponential Growth

Exponential growth is a powerful concept, especially in financial contexts. It refers to situations where quantities increase at a consistently rapid rate over time, typically in proportion to their current value. Let's break it down: imagine you're planting a seed and it's doubling its size every day. That's an example of exponential growth. In financial mathematics, this means your money can grow quite quickly if you invest it wisely. In the context of our exercise, the value of the investment grows exponentially because of continuous compounding. Continuous compounding means the interest calculated isn't just applied once over an entire term but is constantly recalculated, leading to growth on top of growth. This results in significantly faster increases than simple compounding. When you see formulas involving exponential terms like in our problem, the power of exponential growth is demonstrated.

Interest Rate Calculation

Interest rate calculation is the foundation of understanding your gains from an investment. The rate you've negotiated determines just how fast your investment grows. In our exercise, the account earns 6.25% interest, which is part of the reason for the impressive growth over 25 years. To work with interest rates in real-world math scenarios, converting percentages to decimals is crucial. Here, 6.25% becomes 0.0625, which is easier to plug into formulas. Various rates can apply depending on the method of compounding:

• Annually: calculated once a year
• Quarterly: calculated four times a year
• Continuously: recalculated constantly

The exercise employs continuous compounding, which is a mathematical idealization that assumes the compounding frequency is theoretically infinite. This is the most efficient way to harness the interest rate.

Financial Mathematics

Financial mathematics encompasses all tools and formulas that help us understand and maximize our investments.In practical terms, it's about knowing how to calculate future values, assess risks, and determine potential returns.The problem exercise is a perfect example of this. We apply the continuous compounding formula: \( A = Pe^{rt} \) where:

• \( P \) is the principal amount or initial deposit
• \( r \) is the interest rate in decimal form
• \( t \) is the time span in years
• \( e \) is Euler's number (approximately 2.71828)

By understanding these variables and how they interact, you're able to predict the future value of investments accurately.Financial mathematics is critical for making informed decisions, allowing investors to optimize their portfolios, ensuring the maximum return while minimizing risk.