Problem 68 - Algebra and Trigonometry

Question

Compound Interest \(\quad\) A man invests \(\$ 6500\) in an account that pays 6\(\%\) interest per year, compounded continuously. (a) What is the amount after 2 years? (b) How long will it take for the amount to be \(\$ 8000 ?\)

Step-by-Step Solution

Answer

(a) $7338.75; (b)$ about 3.5 years.

Step 1: Understanding Continuous Compound Interest Formula

The formula for calculating continuous compound interest is given by \( A = Pe^{rt} \), where \( A \) is the amount after time \( t \), \( P \) is the principal amount, \( r \) is the annual interest rate, and \( t \) is the time in years. In this problem, we're given \( P = 6500 \), \( r = 0.06 \), and we need to calculate \( A \) for \( t=2 \) years for part (a) and solve for \( t \) when \( A = 8000 \) for part (b).

Step 2: Calculating Amount After 2 Years

To find the amount after 2 years, substitute the known values into the formula: \( A = 6500 \times e^{0.06 \times 2} \). Calculate \( e^{0.12} \) using a calculator, and then multiply by 6500. This computation will give the amount \( A \) after 2 years with continuous compounding.

Step 3: Calculation for Part (a)

Using a calculator, compute \( e^{0.12} \approx 1.1275 \). Then, multiply \( 6500 \times 1.1275 \approx 7338.75 \). So, the amount after 2 years is approximately \$7338.75.

Step 4: Setting up the Equation for Part (b)

For part (b), we need to find the time \( t \) when the amount becomes \$8000. Set up the equation \( 8000 = 6500e^{0.06t} \). Solve for \( t \) by first isolating the exponential term: \( e^{0.06t} = \frac{8000}{6500} \approx 1.2308 \).

Step 5: Solving for Time (t)

Take the natural logarithm on both sides to solve for \( t \): \( 0.06t = \ln(1.2308) \). Calculate \( \ln(1.2308) \approx 0.2071 \). Then, divide by 0.06 to find \( t \): \( t = \frac{0.2071}{0.06} \approx 3.4517 \).

Step 6: Rounding Time to Nearest Year

The calculated \( t \) is approximately 3.4517 years. Since the time is typically expressed in whole years, you can round it to about 3.5 years, or in practical terms, slightly over 3 years and 5 months.

Key Concepts

Continuous Compound Interest Formula

When you invest money in an account with continuous compounding, the interest is constantly added to the principal balance. Thus, you earn interest on interest. The formula for continuous compound interest is:

• \( A = Pe^{rt} \),

where:

• \( A \) is the final amount you will have after a certain time.
• \( P \) is the initial principal (the amount of money you started with).
• \( r \) is the annual interest rate in decimal form.
• \( t \) is the time in years.

For example, if you invest $6500 at a 6% interest rate for 2 years, the formula becomes:

• \( A = 6500e^{0.06 \times 2} \).

This conceptual understanding allows you to calculate both the future value of an investment and the time it takes to reach a certain value.

Exponential Growth

Exponential growth describes a process like continuous compound interest where the value grows by a consistent rate over time. The growth becomes very fast because of the compounding effect, which builds on itself.Exponential growth is a key idea in finance, biology, and even in populations:

• It implies that the larger something gets, the faster it keeps growing.
• The same percentage increase results in much larger absolute increases each time.

In our compound interest problem, the investment grows exponentially due to the formula:

• \( A = 6500e^{0.06t} \).

Every year, you're earning interest on a slightly larger amount, which leads to rapid growth over time.

Natural Logarithm

The natural logarithm, denoted as \( \ln \), is the inverse function of exponential functions. It helps to solve for time in problems of exponential growth, such as continuously compounding interest.When you need to find out how long it will take for your investment to reach a certain value, you'll often rearrange the compound interest formula:

• Take the natural logarithm to isolate the time variable.
• For part (b) of the exercise, set the equation \( 8000 = 6500e^{0.06t} \).

After isolating the exponential expression, you have:

• \( e^{0.06t} = \frac{8000}{6500} \approx 1.2308 \).
• Then apply \( \ln \) to both sides: \( \ln(1.2308) \approx 0.2071 \).

This result is used to solve for \( t \).

Interest Rate Calculation

Understanding interest rate calculation is crucial for estimating how much your money can grow over time with investments. The annual interest rate, represented as \( r \), directly influences the exponential growth of the invested amount.To convert a percentage to a decimal, simply divide by 100. For example, 6% becomes 0.06. This is your rate in the continuous compound interest formula:

• In \( A = Pe^{rt} \), the \( 0.06 \) means that every year, the principal grows by 6% through growth compounded at every moment.

When solving problems, rechecking your interest rate calculation can ensure accuracy in your results. Whether calculating the future value or finding the time period to reach a specific amount, correct interest rate usage is vital.