Problem 144 - 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 decimal places. $$r=6 \%$$

Step-by-Step Solution

Answer

The time required for the investment to double is approximately \(11.55\) years and to triple is approximately \(18.28\) years.

Step 1: Convert Interest Rate

First, convert the interest rate from percentage to decimal form. This can be done by dividing the percentage rate by 100. So, \(r = \frac{6}{100} = 0.06\).

Step 2: Solve For Doubling

Next, plug the values into the continuous compound interest formula and solve for time when the investment doubles. For doubling, \(A = 2P\). Replace \(A\) and \(P\) with 2000 and 1000 respectively. Then, rearrange the formula to solve for \(t\). This would be \[2 = e^{0.06t} \]Take the natural logarithm on both sides of the equation to isolate \(t\), giving us: \[t = \frac{ln(2)}{0.06} \]Calculate \(t\) and round your result to two decimal places.

Step 3: Solve For Tripling

For the case where the investment triples, apply the same process. The only difference is with regards to value of \(A\), it becomes \(3P\), hence the compound interest formula becomes: \[3 = e^{0.06t} \] Then, take the natural log on both sides and solve for \(t\). \[t = \frac{ln(3)}{0.06} \]Calculate \(t\) and round your result to two decimal places.

Key Concepts

Investment Growth

When thinking about investment growth, we often focus on how much our money can increase over time. With continuous compounding, your initial investment grows at every possible moment. This concept allows your money to grow faster compared to other compounding methods.
Continuous compounding uses the formula:

• \( A = Pe^{rt} \)

Here, \( A \) is the final amount, \( P \) is the principal amount (initial investment), \( r \) is the interest rate in decimal form, and \( t \) is the time in years. The constant \( e \) (approximately 2.718) is the base of the natural logarithm. In this case, your money could double or triple, depending on the time and interest rate.
The benefit of understanding this formula is important to calculate how long it will take to achieve certain financial goals.

Natural Logarithm

The natural logarithm, often denoted as \( \ln \), is a special type of logarithm that uses the number \( e \) as the base. It's a fundamental concept used to solve equations involving exponential growth.

• The natural logarithm \( \ln(x) \) tells us the power to which we need to raise \( e \) to get \( x \).

In the context of continuous compound interest, the natural logarithm helps isolate the variable \( t \) in equations like \( A = Pe^{rt} \). By taking the natural log of both sides, we can simplify and solve for \( t \) easily:

• \( \ln(A/P) = rt \)
• \( t = \frac{\ln(A/P)}{r} \)

Understanding this process allows us to determine how long a particular investment needs to grow to reach a specific future value.

Interest Rate

Interest rates have a huge impact on investment growth. They determine how fast or slow your money grows. In continuous compounding, the interest rate \( r \) is used in its decimal form.

• To convert a percentage to decimal, divide by 100. For example, 6% becomes 0.06.

The rate \( r \) represents how much interest accumulates continuously over time. A higher interest rate means your investment grows quickly, while a lower rate results in slower growth.
When analyzing financial plans, it’s important to be mindful of interest rates to make informed decisions. This way, you can project future values more accurately and choose the best investment opportunities.

Time Calculation

Time calculation involves determining how long it takes for an investment to grow to a certain amount using the continuous compounding formula.

• To find \( t \), rearrange the formula: \( t = \frac{\ln(A/P)}{r} \).

If your initial investment doubles, then \( A = 2P \). If it triples, \( A = 3P \). By substituting \( A \) and \( P \) into the formula, you can calculate \( t \) for any scenario.
This equation highlights the relationship between time, interest rate, and the growth of investments. Being able to accurately calculate \( t \) helps investors set realistic goals and expectations. Also, understand how quickly their money can reach these goals, or if adjustments to the interest rate or investments need to be made.