Problem 146 - Algebra and Trigonometry Real Mathematics, Real People

Question

Find the time required for a \(\$ 1000\) investment to (a) double at interest rate \(r,\) compounded continuously, and (b) triple at interest rate \(r\), compounded continuously. Round your results to two $$r=3.75 \%$$

Step-by-Step Solution

Answer

(a) The time required for the investment to double is approximately \(t = 18.44 \) years. (b) The time required for the investment to triple is approximately \( t = 30.14 \) years.

Step 1: Set up the formula for continuous compounding

The formula for continuous compounding is \( A = P e^{rt} \), where \( A \) is the final amount, \( P \) is the initial principal, \( r \) is the annual interest rate in decimal form and \( t \) is the time in years. Here \( P = 1000 \), \( r = 0.0375 \).

Step 2: Solve for (a): time to double the investment

Since the investment should double, \( A = 2P \). Substitute \( A = 2000 \), \( P = 1000 \), \( r = 0.0375 \) into the formula, then solve for \( t \): \( 2000 = 1000 e^{0.0375t} \). This simplifies to \( 2 = e^{0.0375t} \). Taking natural log of both sides, get \( ln(2) = 0.0375t \), then solve for \( t \): \( t = ln(2) / 0.0375 \).

Step 3: Solve for (b): time to triple the investment

Since the investment should triple, \( A = 3P \). Substitute \( A = 3000 \), \( P = 1000 \), \( r = 0.0375 \) into the formula, then solve for \( t \): \( 3000 = 1000 e^{0.0375t} \). This simplifies to \( 3 = e^{0.0375t} \). Taking natural log of both sides, get \( ln(3) = 0.0375t \), then solve for \( t \): \( t = ln(3) / 0.0375 \).

Key Concepts

Interest Rate

Interest rate is the percentage at which your money grows in an investment over a period. It's crucial because it indicates how much profit or return you can expect. There are different ways to apply interest rates, such as simple and compound interest.

• Simple Interest: This is calculated on the principal amount solely. For example, if you invest \(1000 at a 5% interest rate annually, you gain \)50 each year.
• Compound Interest: This is more common as it accounts not only for the initial amount but also includes interest on interest from previous periods. Compounded continuously means that interest is added to the principal for every moment of time.

Using the formula for continuous compounding, \(A = Pe^{rt}\), where:

• \(A\) is the amount after time \(t\)
• \(P\) is the principal amount
• \(r\) is the interest rate as a decimal
• \(t\) is time in years

The interest rate's small changes significantly affect how quickly your investments grow. Understanding this can help you maximize returns.

Natural Logarithm

The natural logarithm, often denoted as \(\ln\), is a fundamental concept in mathematics, especially useful in financial calculations involving exponential growth or decay.

• Relation to Exponentials: The natural log is the inverse of the exponential function. If \(e^x = y\), then \(\ln(y) = x\).
• Base \(e\): Its base, \(e\), is approximately 2.71828 and is important in continuous growth processes.

In our problem, the natural logarithm is used to find out how long it takes for an investment to reach a certain amount.
For example, for doubling an investment: \[2 = e^{0.0375t}\]Taking the natural log of both sides gives: \[\ln(2) = 0.0375t\].
This allows us to solve for \(t\) by dividing \(\ln(2)\) by \(0.0375\). This simplified approach is crucial because complex exponential growth problems are difficult to solve without using natural logs.

Investment Doubling

Doubling an investment refers to reaching a financial amount that is twice the original investment. It's an exciting milestone reflecting effective growth over time. The time it takes for an investment to double depends significantly on the interest rate and its compounding frequency.

• Rule of 72: A handy mental shortcut for estimating how many years it will take to double your money at a compounded annual interest rate. Simply divide 72 by the interest rate percentage (e.g., at an interest rate of 3%, it takes about 24 years).
• Continuous Compounding Calculation: Using the formula \(A = Pe^{rt}\), when \(A = 2P\), solving for \(t\) involves finding \(\ln(2) / r\).

These concepts help investors understand how different factors influence the time required for an investment to double. This insight can guide better investment decisions and optimize financial growth.