Problem 71 - Single Variable Calculus

Question

Find the accumulated amount after 10 years on an investment of \(\$ 10,000\) earning interest at the rate of \(12 \%\) per year compounded continuously.

Step-by-Step Solution

Answer

The accumulated amount after 10 years on an investment of $\$10,000$ earning interest at the rate of $12\%$ per year compounded continuously is approximately $\$33,201.20$.

Step 1: Convert the interest rate to a decimal

To convert the annual interest rate from a percentage to a decimal, we will divide it by 100: \[r = \frac{12}{100} = 0.12\]

Step 2: Identify the given values

Now, let's identify the given values in the problem: \[P = \$10,000 \ , \ r = 0.12 \ , \ t = 10\]

Step 3: Insert values into the formula

Plug the given values into the formula \(A = P e^{rt}\) to calculate the accumulated amount after 10 years: \[A = 10,000 \times e^{(0.12)(10)}\]

Step 4: Calculate the accumulated amount

Use a calculator to compute the accumulated amount: \[A = 10,000 \times e^{1.2} \approx 10,000 \times 3.32012 \approx \$ 33,201.20\]

Step 5: State the answer

The accumulated amount after 10 years on an investment of \(10,000\) earning interest at the rate of \(12 \%\) per year compounded continuously is approximately \(\$33,201.20\).

Key Concepts

Exponential Growth

Exponential growth is a pattern of data that shows greater increases over time, creating a curve on a graph that resembles an ever-steepening slope. In finance, this concept is crucial when dealing with compound interest where money grows at a rate that increases over time because the interest earned is continually added to the principal. This results in the amount accumulating at an increasing rate, much like how populations grow or how a virus may spread exponentially under the right conditions.

When interest is compounded continuously, as in the exercise, the formula for calculating the final amount includes the mathematical constant e, approximately equal to 2.71828. This number is the base of natural logarithms and emerges naturally in various branches of mathematics, especially in calculus when dealing with continuous growth processes.

Compound Interest Formula

The formula for compound interest is pivotal to understanding how investments grow over time. For continuous compounding, the formula takes the form \( A = Pe^{rt} \), where \( A \) represents the accumulated amount after a given time period, \( P \) is the principal amount (initial investment), \( r \) is the interest rate in decimal form, \( t \) is the time the money is invested for, and e is the base of the natural logarithm. In the given exercise, this formula is used to calculate the future value of an investment considering a continuous compounding of interest.

Calculating Accumulated Amount

To calculate the accumulated amount \( A \) using continuous compounding, substitute the values for the principal \( P = $10,000 \), the rate \( r = 0.12 \) (after converting the percentage to decimal form by dividing by 100), and time \( t = 10 \) into the compound interest formula. The exponential component \( e^{rt} \) calculates how much the investment will grow after compound interest is applied continuously over the 10 years. By using a calculator to evaluate \( e^{1.2} \), the result then multiplied by the principal amount gives the accumulated value of the investment after the time period.

Converting Interest Rates

Interest rates are often presented as percentages, which need to be converted into decimal form before using them in financial formulas. This is a fundamental step in properly applying the compound interest formula. Convert the annual interest rate given as a percentage to a decimal by dividing it by 100. For instance, a rate of \( 12\text{%} \) becomes \( 0.12 \) when converted. Doing so aligns with the mathematical syntax of the formulas and ensures accurate calculations of the accumulated amount after factoring in interest rates over time.