Problem 65 - Precalculus : Building Concepts and Connections

Question

Determine how long it takes for the given investment to double if \(r\) is the interest rate and the interest is compounded continuously. Assume that no withdrawals or further deposits are made. Initial amount: \(\$ 2700 ; r=7.5 \%\)

Step-by-Step Solution

Answer

After the calculation, we find that it takes approximately 9.24 years for the initial investment to double when the interest is compounded continuously at an rate of 7.5%.

Step 1: Transform Interest Rate to Decimal

The interest rate is given as 7.5%, but for the formula of the continuous compounded interest, we need to convert this to a decimal. This can be done by dividing the percent by 100: \(r = 7.5 \% = 0.075\).

Step 2: Set Up The Equation

According to the formula of continuously compounded interest, we have \(A = P e^{rt}\). We substitute A by 2P, because the initial investment needs to be doubled. This gives \(2P = P e^{rt}\). We can simplify this equation by dividing both sides by P. This results in the equation \(2 = e^{rt}\).

Step 3: Solve For Time t

In order to solve for time t in the equation \(2 = e^{rt}\), we can apply the natural logarithm (ln) to both sides of the equation. This yields \(ln(2) = ln(e^{rt})\). Using the rule of logarithms, this simplifies to \(ln(2) = rt\). We then divide by r to solve for t: \(t = ln(2) / r = ln(2) / 0.075\).

Step 4: Calculate the Time

Now we substitute the values and calculate the time. Taking the natural logarithm of 2 equals approximately 0.6931. As such, the calculation becomes \(t = 0.6931 / 0.075\).

Key Concepts

Exponential Growth

Exponential growth occurs when the rate of increase of a quantity is proportional to its current size, resulting in the quantity growing at a faster and faster rate over time. This concept is fundamental in understanding compound interest, where your investment grows not just on the initial amount but also on the accumulated interest.

In mathematical terms, this is represented by the formula:

• \[ A = P e^{rt} \]

where:

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

The beauty of exponential growth in the context of compound interest is that it allows an investment to grow faster as it earns interest on both the principal and the accumulated interest over time.

Interest Rate Conversion

Converting an interest rate from percentage to a decimal is an essential step in the process of calculating exponential growth under continuous compounding. This conversion is crucial because mathematical formulas, particularly those involving exponential functions, require the rate to be expressed as a decimal.

For instance, if an interest rate is given as 7.5%, you convert this to a decimal by dividing by 100:

• \[ r = \frac{7.5}{100} = 0.075 \]

This adjusted form of the interest rate allows for straightforward integration into the exponential growth formula, facilitating calculations involving continuous compounding of interest.

Natural Logarithms

Natural logarithms, often denoted as \(ln\), play a vital role in solving equations that involve exponential growth, especially when finding the time it takes for an investment to reach a certain value. The natural logarithm has a base of \(e\), which is the constant approximately equal to 2.71828.

To solve for time in equations like:

• \[ 2 = e^{rt} \]

you use natural logarithms to "undo" the exponential function:

• \[ ln(2) = ln(e^{rt}) \]

According to the properties of logarithms, \(ln(e^{rt})\) simplifies to \(rt\). Hence, you can solve for time \(t\) by dividing both sides by \(r\):

• \[ t = \frac{ln(2)}{r} \]

In this context, natural logarithms allow you to determine the investment's doubling time accurately.

Investment Doubling Time

The investment doubling time is a measure of how long it takes for an initial investment to double in size under a given interest rate with continuous compounding. Calculating this requires understanding of exponential growth, interest rate conversion, and the use of natural logarithms.

Doubling time can be particularly enticing since it gives investors insights into how quickly their money can grow. This calculation involves setting the future value of the investment to twice the initial principal in the exponential growth formula:

• \[ 2P = Pe^{rt} \]

When simplified:

• \[ 2 = e^{rt} \]

The next step involves using natural logarithms to solve for \(t\), leading to the natural logarithm of 2 being divided by the rate:

• \[ t = \frac{ln(2)}{r} \]

This straightforward calculation reveals how effectively your investment grows under continuous compounding, using the natural growth rate of \(e\). Knowing the doubling time helps in planning long-term financial strategies and goals.