Problem 29 - Calculus Early Transcendentals: Pearson New International Edition

Question

How long does it take money to double in value for the specified interest rate? (a) \(6 \%\) compounded monthly (b) \(6 \%\) compounded continuously

Step-by-Step Solution

Answer

(a) About 11.58 years, (b) About 11.55 years.

Step 1: Identify the Formula for Compounded Monthly

To find the time it takes for an investment to double with monthly compounding interest, use the formula \(A = P(1 + \frac{r}{n})^{nt}\), where \(A\) is the future amount, \(P\) is the initial principal balance, \(r\) is the annual interest rate, \(n\) is the number of times the interest is compounded per year, and \(t\) is the number of years. Here, \(A = 2P\), \(r = 0.06\), and \(n = 12\).

Step 2: Set Up the Equation for Monthly Compounding

Since we want the amount to double, set \(A = 2P\). Substituting into the formula gives us \(2P = P(1 + \frac{0.06}{12})^{12t}\). Simplifying, we get \(2 = (1 + \frac{0.06}{12})^{12t}\).

Step 3: Solve for Time with Monthly Compounding

Solve the equation \(2 = (1 + \frac{0.06}{12})^{12t}\) by taking the natural logarithm of both sides: \( \ln(2) = 12t \cdot \ln(1 + \frac{0.06}{12})\). Divide both sides by \(12 \cdot \ln(1 + \frac{0.06}{12})\) to get \(t = \frac{\ln(2)}{12 \cdot \ln(1 + \frac{0.06}{12})}\). Calculate to find \(t \approx 11.58\) years.

Step 4: Identify the Formula for Compounded Continuously

For continuous compounding, use the formula \(A = Pe^{rt}\), where \(e\) is the base of the natural logarithm. Set \(A = 2P\) for doubling, \(r = 0.06\).

Step 5: Set Up the Equation for Continuous Compounding

Since \(A = 2P\), substitute into the formula to get \(2P = Pe^{0.06t}\). Simplify to \(2 = e^{0.06t}\).

Step 6: Solve for Time with Continuous Compounding

Solve \(2 = e^{0.06t}\) by taking the natural logarithm of both sides: \(\ln(2) = 0.06t\). Divide both sides by \(0.06\) to find \(t = \frac{\ln(2)}{0.06}\). Calculate to find \(t \approx 11.55\) years.

Key Concepts

Compounded Interest

When it comes to investing money, understanding "compounded interest" is crucial. Compounded interest means that the interest earned on an investment is calculated on the initial principal and any interest earned previously. For instance, if you invest $100 at a 6% annual interest rate, after one year you would earn $6. In the second year, you don't just earn interest on your original $100, but also on your new total of $106.
This leads to exponential growth, where the investment amount grows faster over time than with simple interest.

• Interest is added at specific intervals (e.g., monthly, yearly).
• It can significantly increase the value of an investment over the long term.

Natural Logarithm

The "natural logarithm" is closely related to compound interest. It is the logarithm to the base of a mathematical constant called "e" (approximately 2.718). The natural logarithm (often represented as "ln") is used to solve for variables in exponential equations, like those involved in calculating compounded interest.

• "e" is a fundamental number in mathematics, much like \(\pi\).
• Natural logarithms simplify the process of solving growth equations.

Suppose you have an equation like \(e^{x} = a\). To find \(x\), you'd need the natural logarithm: \(x = \ln(a)\). This becomes especially useful in financial calculations involving continuous compounding.

Investment Growth

"Investment growth" refers to how an investment increases in value over time, thanks to compounded interest. When calculating how long it takes for an investment to double, compound interest is a key player. Let's explore some factors influencing investment growth:

• Compounding Frequency: The more frequently interest is added, the faster money grows.
• Interest Rate: Higher rates can lead to quicker growth, but it's essential to balance risk and return.
• Time: The longer the investment period, the more opportunity for growth.

By understanding these factors, you can make more informed decisions about investments, potentially maximizing their growth.

Compound Interest Formula

The "compound interest formula" is a cornerstone of finance, enabling individuals to calculate how much their investments or loans will grow or cost. For investments compounded at discrete intervals (like monthly), the formula is:\[ A = P(1 + \frac{r}{n})^{nt} \]Where:

• \(A\) is the future value of the investment.
• \(P\) is the principal amount (initial investment).
• \(r\) is the annual interest rate (as a decimal).
• \(n\) is the number of compounding periods per year.
• \(t\) is the number of years the money is invested.

For continuous compounding, the formula simplifies to:\[ A = Pe^{rt} \]Using these formulas can help answer key financial questions, like how long it takes to double money at a given interest rate.