Problem 72 - 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.. $$\begin{array}{|c|c|c|c|c|c|c|} \hline t & 1 & 10 & 20 & 30 & 40 & 50 \\ \hline A & & & & & & \\ \hline \end{array}$$ $$r=2.5 \%$$

Step-by-Step Solution

Answer

The completed table will have the values of \(A\) for each \(t\): \(1 = \$12,304.35\); \(10 = \$15,828.03\); \(20 = \$20,368.39\); \(30 = \$26,235.01\); \(40 = \$33,822.96\); \(50 = \$43,593.09\).

Step 1: Convert the percentage to a decimal

First, convert the given rate from percentage to decimal. The rate \(r\) is \(2.5\% = 0.025\) in decimal form.

Step 2: Substitute Values into the Exponential Growth Formula

Next, substitute \(P = \$12,000\), \(r = 0.025\), and the various \(t\) values from the table into the exponential growth formula \(A = P e^{rt}\). Calculate \(A\) for each \(t\). For example, for \(t=1\), \(A = \$12,000 e^{0.025 \cdot 1}\)

Step 3: Calculate the Balance

Calculate the balance \(A\) for each \(t\) using a calculator. Repeat step 2 for all \(t\) values in your table.

Step 4: Complete the Table

Finally, write your calculated values of \(A\) in the corresponding places in your table.

Key Concepts

Exponential Growth Formula

The exponential growth formula is a powerful mathematical tool used to calculate the future value of an investment. It is particularly important when dealing with continuously compounded interest. The formula is expressed as:
\[ A = P e^{rt} \]
where:

• \(A\) is the amount of money accumulated after a certain time, including interest.
• \(P\) represents the principal amount initially invested.
• \(r\) is the annual interest rate, expressed as a decimal.
• \(t\) stands for the time, in years, that the money is invested for.
• \(e\) is the base of the natural logarithm, approximately equal to 2.71828.

This formula highlights how fast an investment can grow due to continuous compounding, meaning interest is calculated and added to the principal at an infinitely small rate of intervals. Understanding this formula is crucial for calculating the growth of investments and financial planning.

Interest Rate Conversion

Interest rate conversion refers to the process of changing an interest rate presented as a percentage into a decimal, which is typically used in mathematical formulas. In our exercise, we begin by converting the annual interest rate of 2.5%. To do this, simply divide the percentage by 100:
\[ r = \frac{2.5}{100} = 0.025 \]
This decimal form is necessary for plugging into the exponential growth formula. Converting percentages to decimals can make calculations simpler and enables a straightforward input into formulas. It is important to ensure accuracy during this step, as any mistake in conversion might lead to incorrect results in the final calculation.

Investment Calculation

Investment calculation involves determining the future value of an initial investment based on given parameters, such as the principal amount, interest rate, and time period. Using our standardized exponential growth formula, we calculated the future value of an investment of $12,000 at an annual interest rate of 2.5%, compounded continuously. For each time period specified in the table:

• Plug in the given \(P\), \(r\), and \(t\) values into the formula \(A = P e^{rt}\).
• Use a scientific calculator to compute the result for each specified \(t\) (1, 10, 20, 30, 40, and 50 years).
• Update the table with the calculated balances \(A\).

This process helps investors understand how their savings will grow over time, taking continuous compounding into account.

Time Value of Money

The time value of money is a financial concept that posits a sum of money available now is worth more than the same sum in the future due to its potential earning capacity. This principle is embedded in the calculation of continuously compounded interest.
Through continuous compounding, interest is accrued on the initial principal as well as on the accumulated interest from previous periods, making your money grow exponentially over time. In our exercise, examining balance calculations over different time periods, we see the practical application of the time value of money.

• As \(t\) increases, the accumulated value \(A\) becomes significantly larger, showing how money can grow exponentially with time.
• This compounding effect is especially noticeable over long periods, emphasizing the benefits of starting investments early.

Understanding this concept encourages strategic investing and reinforces the idea of letting your money work for you over extended durations.