Problem 71 - Algebra and Trigonometry Real Mathematics, Real People

Question

Complete the table to determine the balance \(A\) for \(\$ 12,000\) invested at rate \(r\) for \(t\) years, compounded continuously.. $$\begin{array}{|c|c|c|c|c|c|c|} \hline t & 1 & 10 & 20 & 30 & 40 & 50 \\ \hline A & & & & & & \\ \hline \end{array}$$ $$r=3.5 \%$$

Step-by-Step Solution

Answer

The balance after 1, 10, 20, 30, 40, 50 years would be approximately $12,420, $16,386, $21,544, $28,374, $37,328, and $49,183 respectively.

Step 1: Identify the variables

Here, the principal \(P\$12000\), interest rate \(r\) is \(3.5\%\) and the time \(t\) is given in the table as 1, 10, 20, 30, 40, 50 years.

Step 2: Convert interest rate to decimal

The rate \(r\) is given as a percentage. To use in formula, convert it to decimal by dividing by 100. Thus, \(r = 3.5\% / 100 = 0.035\).

Step 3: Apply formula and calculate A for each duration

Apply the formula \(A = P e^{rt}\) and calculate \(A\) for each time in the table. So, for \(t= 1\), \(A = 12000 * e^{(0.035 * 1)}\). Repeat this for \(t = 10, 20, 30, 40, 50\) years.

Key Concepts

Continuous Compound Interest

Understanding how continuous compound interest works is crucial for anyone looking to invest or save money. Rather than being compounded at regular intervals, such as monthly or annually, continuous compounding calculates interest accumulation at every possible instant. This concept can be represented by the formula:
\[ A = Pe^{rt} \]
where \( A \) is the future value of the investment, \( P \) is the principal amount, \( r \) is the annual interest rate (in decimal form), and \( t \) is the time the money is invested for, in years. Mathematically, continuous compounding uses the mathematical constant \( e \), which is approximately equal to 2.71828. This number is the base of the natural logarithm and it appears frequently in calculus and mathematical modeling of growth patterns.

Exponential Growth

When an investment experiences continuous compound interest, it exhibits exponential growth. This means the rate of increase becomes quicker over time. Exponential growth is characterized by the presence of the variable in the exponent of the growth function, as seen in the continuous compounding formula.
In the context of the exercise, the exponential factor is \( e^{rt} \), where \( r \) represents the growth rate, and \( t \) is the time. Because this factor grows exponentially with time, even slight differences in the interest rate or the investment duration can lead to significant changes in the final balance. This principle underscores the power of compounding - the earlier the investment is made or the higher the interest rate, the more pronounced the exponential growth will be.

Time Value of Money

The time value of money is a financial concept which states that a dollar today is worth more than a dollar in the future due to its potential earning capacity. This core principle underpins the notion of interest and investment returns. It quantifies the opportunity cost of waiting for future funds. Applying this concept, investors prefer to receive money today rather than the same amount in the future. Continuous compound interest directly relates to this as it maximizes the growth potential of money over time, further emphasizing the significance of starting to invest early.

Calculating Investment Balance

To calculate an investment balance that is being compounded continuously, you can use the formula mentioned previously. Here’s a brief step-by-step for the exercise problem:

• Identify the principal amount (\( P = $12,000 \)), the annual interest rate in decimal form (\( r = 0.035 \)), and the time period (\( t \)).
• Apply the continuous compounding formula for each time period to find the balance \( A \).
• For example, after 1 year, the balance would be calculated as \( A = 12000 * e^{(0.035 * 1)} \).
• Repeat this process for 10, 20, 30, 40, and 50 years to see how the balance grows over time.

Understanding and applying these calculations is key for forecasting the future value of an investment and making informed financial decisions.