Problem 70 - Single Variable Calculus

Question

Find the accumulated amount after 5 years on an investment of \(\$ 5000\) earning interest at the rate of \(10 \%\) per year compounded continuously.

Step-by-Step Solution

Answer

The accumulated amount after 5 years on an investment of \$5000 earning interest at the rate of 10% per year compounded continuously is approximately \$8243.64.

Step 1: Identify the given information

We are given the following information in the problem: - Principal amount (P): \(\$ 5000\) - Interest rate (r): \(10 \%\) - Time (t): 5 years

Step 2: Convert the interest rate into decimal form

To use the interest rate in the continuous compound interest formula, we need to convert the percentage into a decimal format. To do this, simply divide the given percentage by 100: \(r = \frac{10}{100} = 0.10\)

Step 3: Apply the continuous compound interest formula

We will now use the continuous compound interest formula: \(A = P \cdot e^{rt}\) Plugging in the given values: \(A = 5000 \cdot e^{(0.10)(5)}\)

Step 4: Calculate the accumulated amount

Now, calculate the accumulated amount by using a calculator to find the value: \(A = 5000 \cdot e^{(0.10)(5)} = 5000 \cdot e^{0.50} \approx 8243.64\) The accumulated amount after 5 years on an investment of \(5000 earning interest at the rate of 10% per year compounded continuously is approximately \)\$ 8243.64$.

Key Concepts

Exponential Growth

Exponential growth is a fundamental concept in various fields, including finance. In terms of investments, it describes how money grows over time at a constant rate. This happens when the amount of money increases rapidly and its growth accelerates as the value increases.

For continuous compound interest, this growth follows an exponential pattern. The formula involved is: - \[ A = P \, e^{rt} \] - Here, \( A \) is the accumulated amount, \( P \) is the principal, \( e \) is the base of the natural logarithm (approximately 2.71828), \( r \) is the interest rate as a decimal, and \( t \) is time in years.

Exponential growth is powerful because even small increases in \( r \) or \( t \) can significantly impact \( A \). This makes it a crucial concept for understanding how investments grow.

Interest Rate

The interest rate is a key factor in determining how much investment will grow over time. It represents the proportion of the principal amount that is paid as interest over a specified period.

Interest rates can be stated as annual percentage rates (APR). In the context of continuous compounding, like in the original exercise, it is necessary to convert this percentage into a decimal. This is done by dividing by 100. For example, a 10% interest rate becomes 0.10.

Using the decimal form helps when plugging values into the continuous compound interest formula. The rate at which your investment grows depends directly on the interest rate, which makes understanding this conversion important.

• Higher interest rates mean faster growth.
• Lower interest rates mean slower growth.

Principal Amount

The principal amount in the context of investments is the initial sum of money invested or borrowed. This is represented by \( P \) in the continuous compound interest formula.

It's the starting point for any investment calculation. In our exercise, the principal amount is \( \$5000 \). Understanding the principal is simple: it is the base amount that will earn interest over time.

The principal directly affects the accumulated amount. Larger principals will potentially result in more accumulated interest, assuming all other factors are equal. Hence, knowing your principal helps set realistic financial goals.

• Starting with a larger principal increases potential earnings.
• A smaller principal might limit initial growth but still benefits from compounding.

Investment Calculation

Investment calculation through continuous compounding is effective for predicting future growth. Here, we use the formula \( A = P \, e^{rt} \). Here’s a breakdown of how it works in practical terms:

1. Start with your initial investment, or principal \( P \).
2. Know your annual interest rate \( r \), in decimal form.
3. Define your time period \( t \) in years.
4. Using these inputs in the formula provides the accumulated amount \( A \).

In our exercise, substituting \( P = 5000 \), \( r = 0.10 \), and \( t = 5 \), we calculate \( A = 5000 \, e^{0.50} \), which gives approximately \$8243.64.

Investment calculations are crucial for financial planning, ensuring that investors can anticipate their returns over time. Adjusting variables like the principal, rate, or time allows different scenarios to be analyzed easily.