Problem 17 - Calculus and its applications

Question

Complete the following. $$ \begin{array}{lcl} \text { Initial } & {\text { Doubling }} \\ \begin{array}{l} \text { Investment } \\ \text { at } t=0, P_{0} \end{array} & \begin{array}{l} \text { Interest } \\ \text { Rate, } k \end{array} & \begin{array}{l} \text { Time, } T \\ \text { (in years) } \end{array} & \begin{array}{l} \text { Amount } \\ \text { after } 5 \mathrm{yr} \end{array} \\ \hline \$ 75,000 & 6.2 \% & & \\ \hline \end{array} $$

Step-by-Step Solution

Answer

The investment doubles in about 11.61 years, and the amount after 5 years is approximately $101,309.

Step 1: Understanding the Problem

You are given an initial investment of $75,000 and an interest rate of 6.2% per year. You need to calculate the time, \( T \), at which the investment will double and the amount after 5 years. The formula to find the doubling time using compound interest is the Rule of 72: \( T \approx \frac{72}{k} \).

Step 2: Apply the Rule of 72

To find the doubling time \( T \), use the approximation \( T \approx \frac{72}{k} \) where \( k = 6.2 \). Calculate \( T \) as follows: \( T \approx \frac{72}{6.2} \).

Step 3: Calculate the Doubling Time

Perform the division: \( T \approx \frac{72}{6.2} \approx 11.61 \). Therefore, the investment will double in approximately 11.61 years.

Step 4: Determine the Amount after 5 Years

Use the compound interest formula \( A = P_0(1 + \frac{k}{100})^t \), where \( t = 5 \) years and the interest rate \( k = 6.2 \% \). Calculate the amount: \( A = 75000 \times (1 + \frac{6.2}{100})^5 \).

Step 5: Calculate the Final Amount

Perform the calculation: first, find \( (1 + \frac{6.2}{100}) \approx 1.062 \). Then compute \( 1.062^5 \approx 1.35078 \). Finally, \( A = 75000 \times 1.35078 \approx 101,309 \).

Key Concepts

Doubling Time

The concept of doubling time is all about figuring out how long it takes for an investment to double in value at a given interest rate. Imagine planting a money seed—how quickly does it become a money tree with twice as many bills as before? To find when your investment doubles, you can use a neat trick called the "Rule of 72." This rule gives you a quick way to estimate how many years it will take.

• Doubling Time Formula: The quick formula is: \( T \approx \frac{72}{k} \), where \( T \) is the doubling time, and \( k \) is the interest rate expressed as a percentage.
• Example with 6.2% interest: To find out when the $75,000 investment doubles at a rate of 6.2%, you use the formula: \( T \approx \frac{72}{6.2} \). This gives approximately 11.61 years.

This rule works well for interest rates that are not too high or too low, and it's a handy shortcut whenever you're curious about doubling times.

Rule of 72

The Rule of 72 is a simple yet powerful tool used in finance to quickly estimate the doubling time of an investment. This rule helps you avoid complex calculations. All you need is the interest rate, and you can find out how long it takes for your money to grow.

• What It Is: The rule states that to find the doubling time, divide 72 by your interest rate.
• Why 72? The number 72 is a result of logarithmic calculations, and it serves as an easy-to-remember figure that approximates the doubling time in years.
• Quick Example: With a 6.2% interest rate, using the Rule of 72, it takes \( \frac{72}{6.2} \approx 11.61 \) years for money to double.

One key advantage of the Rule of 72 is its simplicity and speed, making it a favorite among financial planners and investors for quick assessments.

Investment Growth

Investment growth refers to how much your investment will increase over time, which depends heavily on the interest rate and the duration of the investment. It's exciting to see how your initial investment blossoms over the years.

• Compound Interest Formula: Use the formula \( A = P_0(1 + \frac{k}{100})^t \) to find out the amount you will have after a certain time \( t \) years.
• Example Calculation: For \(75,000 at 6.2% over 5 years, you calculate \( A = 75000 \times (1 + \frac{6.2}{100})^5 \).
• Results: This product gives \( A \approx 75,000 \times 1.35078 \approx 101,309 \). Thus, after 5 years, the investment grows to approximately \)101,309.

This growth is why investing can be so rewarding. Watching your money multiply over time is one of the key benefits of compound interest.