Problem 50 - Algebra A Combined Function

Question

Use the formula \(A=P e^{r t}\) to solve. Find the amount of money for which a \(\$ 2500\) certificate of deposit is redeemable if it has been earning \(10 \%\) interest compounded continuously for 3 years.

Step-by-Step Solution

Answer

The amount is approximately $3374.75.

Step 1: Identify the Variables

The formula for continuously compounded interest is given as \(A = Pe^{rt}\), where \(A\) is the amount after time \(t\), \(P\) is the principal amount, \(r\) is the annual interest rate, and \(t\) is the time in years. From the problem, \(P = 2500\), \(r = 0.10\), and \(t = 3\).

Step 2: Substitute the Values into the Formula

Substitute \(P = 2500\), \(r = 0.10\), and \(t = 3\) into the formula: \[ A = 2500 \times e^{0.10 \times 3} \]

Step 3: Calculate the Exponent

Calculate \(0.10 \times 3\): \[0.10 \times 3 = 0.3\] So the formula becomes: \(A = 2500 \times e^{0.3}\).

Step 4: Compute the Exponential Value

Use a calculator to find \(e^{0.3}\). \[e^{0.3} \approx 1.3499\] Substituting this back into the formula gives: \[A = 2500 \times 1.3499\]

Step 5: Calculate the Final Amount

Multiply the principal \(2500\) by the exponential value \(1.3499\) to find \(A\): \[A = 2500 \times 1.3499 \approx 3374.75\]

Step 6: Interpret the Result

The amount of money for which the \(\\(2500\) certificate of deposit is redeemable after 3 years, with continuous compounding at a \(10\%\) rate, is approximately \(\\)3374.75\).

Key Concepts

compound interest

Compound interest is a financial concept where you earn interest on your initial amount, known as the principal, as well as on any interest that accumulates over time. This differs from simple interest, which is only calculated on the principal amount.

• With compound interest, your money grows faster compared to simple interest because each year, or compounding period, the interest you earn starts to earn interest itself.
• This creates a snowball effect, known as the compounding effect, where the total amount grows more substantially with each period.

In the given problem, compound interest is applied continuously, meaning that the interest is calculated and added at every possible instant—leading to the use of the constant \( e \), which is approximately equal to 2.71828, in the calculation. To summarize, with continuous compounding, the formula \( A = Pe^{rt} \) allows you to see how your investment grows over time, showcasing the power of compounding.

exponential growth

Exponential growth describes a situation where the rate of growth is proportional to the current size, meaning as something grows, it grows more rapidly. This type of growth is marked by a constant percentage rate over time, forming a J-shaped curve when graphed.

• In the context of finance, exponential growth applies when interest compounds continuously, causing the investment to grow at an increasingly rapid rate.
• This continuous increase is a hallmark of exponential growth, where, as in the problem, calculating \( e^{0.3} \) reveals how an investment of \( \$2500 \) evolves over 3 years.

Understanding exponential growth is important because it helps you better appreciate how investments can increase significantly in value over time with the right conditions, such as a constant interest rate and continuous compounding. Therefore, it emphasizes the importance of time and compound frequency in financial planning.

financial mathematics

Financial mathematics combines mathematical methods with financial theories to solve problems related to investments, like calculating compound interest, loans, and savings. It's a foundational aspect of understanding how money works and how to maximize the return on investments.

• This field relies on formulas, like the continuous compounding formula \( A = Pe^{rt} \), to make informed financial calculations and decisions.
• It provides insights into critical variables affecting finances, like interest rates, time periods, and the principles of growth and decay.

In the given exercise, we see financial mathematics in action as it allows for determining the future value of a \( \$2500 \) deposit growing under continuous compounding. Such calculations are crucial for both personal and professional financial planning, enabling individuals and businesses to forecast growth, assess risks, and optimize their financial strategies.