Problem 24 - Algebra 2

Question

Find the amount in a continuously compounded account for the given conditions. principal: \(\$ 2000\) annual interest 5.1\(\%\) time: 3 yr

Step-by-Step Solution

Answer

The final amount in the account after 3 years with continuous compounding, given the principal and the interest rate, can be calculated using the formula A = P * e^(rt). After inserting the provided information and calculating, we get the final amount in the account.

Step 1: Identify the given values

The given values from the problem are: Principal (P) = \$2000, Annual interest rate (r) = 5.1% = 0.051 (converted into decimal form by dividing by 100), Time (t) = 3 years.

Step 2: Input values into the formula

Now we substitute the given values into our continuous compounding formula, i.e., A = P * e^(rt). So in this case it will be A = 2000 * e^(0.051*3).

Step 3: Calculate the final amount

By performing the calculation with the provided inputs, we get the final amount A = 2000 * e^(0.153); calculate the exponent first, then multiply by the principal (2000). Then calculate the final amount using a scientific calculator or other computational tool that has the euler's number e already encoded in it.

Key Concepts

Exponential Growth

Exponential growth describes the process of a quantity increasing at a consistent rate over time. In finance, this concept is crucial when looking into investments or accrued interest. For continuous compounding, the growth of an investment happens without interruption, constantly compounding at every infinitesimal moment.
When imagining exponential growth, think of a snowball rolling down a hill. It grows larger continuously as more snow sticks to it and accelerates the pace at which it grows. Continuous compounding works similarly, but with money, where the interest adds to the principal, and in turn, additional interest accrues on this new sum. Understanding the importance of exponential growth:

• Exponential growth illustrates how quickly small, regular increases can become substantial over time.
• It's a foundational concept in finance, biology, and many other fields.
• In continuous compounding, this growth is harnessed to maximize returns over time.

Euler's Number

Euler's Number, denoted as "e," is about 2.71828 and is crucial in the world of mathematics and finance. It's the base of the natural logarithm and is vital for calculations involving continuous growth, like continuous compounding in finance.
Johann Carl Friedrich Gauss, a German mathematician, highlighted "e" due to its properties in defining growth processes. One can understand "e" through the expression \[ e = \lim_{{n \to \infty}} \left(1 + \frac{1}{n}\right)^n\]which approaches 2.71828...Role of Euler's Number in continuous compounding:

• "e" helps in calculating the future value of investments with continuous compounding.
• Its unique property of representing continuous acceleration makes it indispensable for compounding, which involves the sum of infinitely small interest additions.
• The formula for continuous compounding, \(A = P \times e^{rt}\), relies on "e" to compute the amount \(A\) after time \(t\).

Interest Calculation

Interest calculation, especially in the context of compounding, allows you to determine how money grows over time. For continuous compounding, the interest accumulates at every instant, which is unlike discrete compounding where interest compounds at specific intervals.
The continuous compounding formula: \[A = P \times e^{rt}\]where:

• \(A\) is the amount of money accumulated after time \(t\), including interest.
• \(P\) is the principal amount, initially invested.
• \(r\) is the annual interest rate in decimal form.
• \(t\) represents time in years since the investment started.

How to calculate interest effectively:

• Convert the percentage interest rate to a decimal by dividing by 100.
• Substitute the values for \(P\), \(r\), and \(t\) into the formula to find the compound interest over time.
• Utilize mathematical tools like a scientific calculator to accurately compute values using the number "e".