Problem 95 - Algebra and Trigonometry with Analytic Geometry

Question

One of the oldest banks in the United States is the Bank of America, founded in 1812. If \(\$ 200\) had been deposited at that time into an account that paid \(4 \%\) annual interest, then 180 years later the amount would have grown to \(200(1.04)^{180}\) dollars. Approximate this amount to the nearest cent.

Step-by-Step Solution

Answer

The approximate amount after 180 years is \(\$209,634.00\).

Step 1: Understand the Formula

The problem provides the formula to calculate the future value of an investment with compound interest: \( A = P(1 + r)^n \). Here, \(P = \$200\) is the principal, \(r = 0.04\) is the annual interest rate, and \(n = 180\) is the number of years.

Step 2: Substitute Known Values

Substitute the given values into the formula: \( A = 200(1.04)^{180} \). Our task is to calculate this expression to find the future value of the deposit.

Step 3: Use a Calculator

To compute \(200(1.04)^{180}\), use a scientific calculator. First, calculate \((1.04)^{180}\), then multiply the result by 200.

Step 4: Calculate (1.04)^{180}

Use a scientific calculator or software to compute \((1.04)^{180}\). This might be a large number, close to approximately 1,048.17.

Step 5: Multiply by the Principal

Now, multiply the result from Step 4 by 200: \(200 \times 1,048.17 \approx 209,634.00\).

Step 6: Round to the Nearest Cent

Since money amounts are typically rounded to two decimal places, the final amount is approximately \(\$209,634.00\).

Key Concepts

Future Value

The future value is a financial concept that helps project the future price of an investment based on a specified rate of return and time period. When dealing with compound interest, this is the amount at which an investment will grow after a set period.
The formula to calculate the future value of an investment is \[ A = P(1 + r)^n \]where:

• \(A\) is the future value.
• \(P\) is the principal amount.
• \(r\) is the annual interest rate.
• \(n\) is the number of years the money is invested or borrowed for.

Through the power of compounding, your money can multiply significantly over time. Compounding means that you earn interest not only on your initial investment but also on the interest accumulated from previous periods.

Annual Interest Rate

The annual interest rate is a key component of investment calculations. It is the percentage of the principal earned or paid per year. This rate determines how much your initial investment will grow or how much a loan will cost over time.
An annual interest rate of 4% means that each year, your investment will grow by an additional 4% of its total value from the previous year. As years pass, you earn interest on an increasingly larger balance, thanks to the magic of compounding. Therefore, even a seemingly small interest rate can lead to substantial gains given sufficient time.

Investment Calculation

Investment calculation involves determining the current value of an investment and predicting its future value using factors like the principal amount, interest rate, and time period. Calculators or financial formulas like the compound interest formula can be used to make these predictions.
The process generally involves:

• Identifying initial investments and expectations.
• Utilizing the appropriate financial formula.
• Computing values using current and historical financial data.

In our example, we calculated the future value of a deposit over 180 years using a 4% annual interest rate and the formula for compound interest. This helps establish expectations and plan financial goals effectively.

Principal Amount

The principal amount is the initial sum of money deposited or borrowed. It is crucial because it is the base amount on which interest is calculated over investment or loan periods.
In our scenario, the principal amount is $200. This is the starting value from which interest accumulates yearly. The choice of principal can significantly impact the future value; larger initial investments tend to yield higher future values, given the same interest rate and time period.
Understanding the principal is foundational when planning investments, as it anchors the investment calculations and future projections.