Problem 31 - Algebra and Trigonometry

Question

Compound Interest If \(\$ 2000\) is invested at an interest rate of 3.5\(\%\) per year, compounded continuously, find the value of the investment after the given number of years. \(\begin{array}{llll}{\text { (a) } 2 \text { years }} & {\text { (b) } 4 \text { years }} & {\text { (c) } 12 \text { years }}\end{array}\)

Step-by-Step Solution

Answer

(a) $2145.02, (b) $2300.54, (c) $3042.10.

Step 1: Understand the Compound Interest Formula

When interest is compounded continuously, we use the formula \( A = Pe^{rt} \), 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 in decimal form, and \( t \) is the time in years. \( e \) is the base of the natural logarithm, approximately equal to 2.71828.

Step 2: Convert the Interest Rate to Decimal

To use the formula, we convert the interest rate of 3.5% to decimal form by dividing by 100. So, \( r = \frac{3.5}{100} = 0.035 \).

Step 3: Calculate the Investment for 2 Years

Substitute \( P = 2000 \), \( r = 0.035 \), and \( t = 2 \) into the formula: \( A = 2000 \times e^{0.035 \times 2} \). Solving this, we find \( A = 2000 \times e^{0.07} \approx 2000 \times 1.07251 \approx 2145.02 \).

Step 4: Calculate the Investment for 4 Years

Substitute \( t = 4 \) into the formula: \( A = 2000 \times e^{0.035 \times 4} \). Solving this, we find \( A = 2000 \times e^{0.14} \approx 2000 \times 1.15027 \approx 2300.54 \).

Step 5: Calculate the Investment for 12 Years

Substitute \( t = 12 \) into the formula: \( A = 2000 \times e^{0.035 \times 12} \). Solving this, we find \( A = 2000 \times e^{0.42} \approx 2000 \times 1.52105 \approx 3042.10 \).

Key Concepts

Continuously Compounded Interest

Continuously compounded interest is a powerful concept that can boost your investment over time. Unlike simple interest, which is calculated using only the original principal, continuously compounded interest takes into account the interest added to the balance continuously.
This means the interest is calculated and added back to the principal at every possible moment. This constant addition becomes a natural exponential growth over time using the formula:

• \[ A = Pe^{rt} \]
• Where \( A \) represents the future value of the investment,
• \( P \) is the principal amount,
• \( r \) is the interest rate converted to decimal form,
• \( t \) is the time the money is invested for in years,
• And \( e \) is Euler's number, approximately 2.71828.

Understanding and using this formula allows you to calculate the future value of an investment with continuous growth. It's particularly beneficial over longer periods.

Investment Growth

Investment growth through the lens of compound interest provides a remarkable opportunity for wealth expansion. With continuous compounding, the key is in how frequently the interest is reinvested. Here, growth occurs at every possible instant, which can make a noticeable difference even with a seemingly small interest rate.
For instance, an initial investment of $2000 growing at a 3.5% annual interest rate becomes significantly larger over time:

• After 2 years, the investment grows to $2145.02.
• In 4 years, it expands to $2300.54.
• By 12 years, it reaches $3042.10.

The principle behind this growth is reinvesting the accrued interest, which results in a compounding effect. This method maximizes the growth potential of an investment over time.

Exponential Growth

Exponential growth is a mathematical concept describing how an amount increases at a consistent relative rate over time. In terms of compounded interest, it encapsulates the snowballing growth of your investment.
The foundation of exponential growth in finance is based on the understanding that each piece of added interest becomes part of the principal for calculating future interest. This is truly a growth form of compounding itself.
Consider continuously compounded interest as a representation of this exponential phenomenon. The growth curve steepens over time, emphasizing how money can multiply significantly even with moderate rates due to:

• The endless compounding of interest,
• The impact of time on enhancing returns,
• The effect of even minor rate changes over the years.

When visualized, the graph of exponential growth is not a straight line but a curve that rises ever upwards.

Interest Rate Conversion

Interest rate conversion is a crucial step in investment calculations, enabling you to work seamlessly with percentages in mathematical formulas. To make use of the continuous compounding formula, the interest rate needs to be in decimal form, not percentage.
Here's how you convert:

• Take the percentage rate and divide by 100
• For example, a 3.5% interest converts to 0.035 by dividing: \( \frac{3.5}{100} \)

This conversion to decimal is essential, as it forms part of the formula for finding the future value of your investment. It helps unify calculations and ensures that rates are expressed in a standard mathematical format. Understanding this conversion is key to executing financial calculations accurately.