Problem 49 - Algebra A Combined Function

Question

Use the formula \(A=P e^{r t}\) to solve. How much money does Barbara Mack owe at the end of 4 years if \(6 \%\) interest is compounded continuously on her \(\$ 2000\) debt?

Step-by-Step Solution

Answer

Barbara Mack owes approximately $2542.50 at the end of 4 years.

Step 1: Identify Given Variables

The variables in the formula \(A=Pe^{rt}\) are as follows: \(P\) is the principal amount, \(r\) is the rate of interest, and \(t\) is the time period. From the question, we identify \(P = 2000\), \(r = 0.06\), and \(t = 4\).

Step 2: Substitute Values into Formula

Substitute the identified values into the formula: \[A = 2000 \times e^{0.06 \times 4}\].

Step 3: Calculate the Exponent

First, calculate the exponent: \(0.06 \times 4 = 0.24\). Thus, the formula becomes \[A = 2000 \times e^{0.24}\].

Step 4: Evaluate the Exponential Component

Calculate \(e^{0.24}\) using a calculator. This results in approximately \(1.27125\).

Step 5: Calculate the Final Amount

Multiply the result by the principal \(P\): \[A = 2000 \times 1.27125 = 2542.50\].

Key Concepts

Exponential Functions

Exponential functions are mathematical expressions where a constant base is raised to a variable exponent. They are characterized by a constant rate of growth or decay. This constant base is often represented by the mathematical constant \(e\), which is approximately equal to 2.71828. In finance and other contexts, \(e\) comes into play in continuous growth scenarios such as compounding interests.
Understanding exponential functions is crucial because they model various growth patterns in real-life situations, such as population growth, radioactive decay, and investment growth through compound interest.
When utilizing exponential functions, remember that the rate at which the function increases is proportional to its current value, which gives them their unique runaway effect.

Continuous Compounding

Continuous compounding refers to the mathematical idea where interest is calculated and added to the principal continuously, allowing interest to be accrued at every possible moment. Unlike traditional compounding, which might occur annually, semi-annually, or quarterly, continuous compounding uses the power of exponential functions to determine how the principal grows over time.
The formula for continuous compounding is typically given by \(A = Pe^{rt}\), where:

• \(A\) is the final amount.
• \(P\) is the initial principal balance.
• \(r\) is the annual interest rate.
• \(t\) is the time in years.

The concept of continuous compounding is essential because it yields a slightly higher return compared to other compounding methods when the same rate is used. This makes it a vital concept for long-term financial planning and scenarios where the utmost precision in interest calculation is required.

Interest Calculation

Interest calculation, especially when compounded continuously, is crucial in understanding how an investment grows over time. To calculate the future value of an investment with continuous compounding, we use the formula \(A = Pe^{rt}\).
Here's how you apply it:

• Start with the initial amount (principal \(P\)) you want to invest.
• Multiply the annual interest rate \(r\) (expressed as a decimal) by the time \(t\) the money is invested.
• Raise \(e\), the base of natural logarithms, to this product.
• Finally, multiply the result by your principal \(P\) to find the accumulated amount \(A\).

This process allows you to determine how much money you can expect after a certain period if your interest is compounded continuously. By understanding all variables involved, you can make informed decisions about savings and investments.

Algebraic Formulas

Algebraic formulas like \(A = Pe^{rt}\) serve as vital tools in solving compound interest problems. With a solid grasp of algebra, you can manipulate and solve these formulas to find unknown variables.
Here’s a breakdown of how the formula components work:

• \(A\) is the desired end value or the amount you want to calculate.
• \(P\) refers to the principal you initially invest.
• \(e\) is the mathematical constant for the natural exponential function.
• \(r\) is the interest rate, always expressed as a decimal.
• \(t\) signifies the time in years.

Understanding how to substitute and solve these variables can help unlock problem-solving abilities not only for financial calculations but across various fields that require precise computational analysis. Algebra is the foundation that allows for the manipulation of these expressions to derive meaningful results from complex data sets.