Problem 39 - Elementary and Intermediate Algebra

Question

Find A using the formula \(A=P e^{r t}\) given the following values of \(P, r,\) and \(t .\) Round to the nearest hundredth. $$ P=565, r=-0.5 \%, t=8 \text { years } $$

Step-by-Step Solution

Answer

The calculated amount \( A \) is approximately \( 542.65 \).

Step 1: Understand the Formula

The formula given is \( A = P e^{rt} \), where \( A \) is the final amount, \( P \) is the principal amount, \( r \) is the interest rate expressed as a decimal, and \( t \) is the time in years. We need to find the value of \( A \) using this formula.

Step 2: Convert Percentage to Decimal

The interest rate \( r \) is given as \(-0.5\%\). To convert it into a decimal, divide by 100: \( r = -0.5 \div 100 = -0.005 \).

Step 3: Calculate the Exponent

Substitute the values of \( r \) and \( t \) into the exponent: \( rt = -0.005 \times 8 = -0.04 \).

Step 4: Compute \( e^{rt} \)

Calculate \( e^{-0.04} \). Using a calculator, \( e^{-0.04} \approx 0.9608 \).

Step 5: Calculate \( A \)

Now substitute \( P = 565 \) and \( e^{rt} \approx 0.9608 \) into the formula: \( A = 565 \times 0.9608 \).

Step 6: Compute the Result

Perform the multiplication: \( 565 \times 0.9608 \approx 542.652 \).

Step 7: Round the Result

Round \( 542.652 \) to the nearest hundredth: \( 542.65 \).

Key Concepts

Continuous Compounding

Continuous compounding refers to the process of earning interest on an investment, where the interest is calculated and added to the principal balance an infinite number of times per year. Quite simply, instead of being compounded monthly or quarterly, your investment grows continuously. This concept is vital because it results in the maximum possible return when the rate of interest gets compounded over a certain period.
The main advantage of continuous compounding is that it allows the investment to grow at a constant rate exponentially. This growth is represented by the exponential function in the formula. In practical terms, the more frequently interest is compounded, the more you earn.
In the context of our exercise, the formula for continuous compounding is represented as \[ A = P e^{rt} \]where:

• A: the amount of money accumulated after n years, including interest.
• P: the principal amount (initial investment).
• r: the annual interest rate (in decimal).
• t: the time the money is invested for in years.

Exponential Functions

An exponential function is a mathematical function of the form \[ f(x) = a imes e^{bx} \]where "e" is the base of natural logarithms, approximately equal to 2.71828, "a" is a constant, and "b" is the rate of growth or decay. In the scope of our exercise, the exponential function represents the growth or decay of an investment over time.
In continuous compounding, the continuous growth factor, \[ e^{rt} \]governs how the principal amount (P) evolves. Here, "r" is the rate, and "t" is the time. The exponent's sign, positive or negative, determines whether the function models growth or decay.
For example:

• Exponential Growth: This occurs when the rate "r" is positive. The principal amount grows over time.
• Exponential Decay: This happens when the rate "r" is negative, as seen in our exercise with \( r = -0.005 \). The principal amount decreases over time.

Understanding exponential functions allows us to describe how processes grow or shrink quickly, such as populations, investments, or radioactive decay.

Algebraic Formulas

Algebraic formulas, particularly in finance, are essential for solving problems related to investments, interests, and loans. Mastery of these formulas enables us to determine future values, present values, interest rates, and time periods for financial scenarios.
The equation \[ A = P e^{rt} \]is an algebraic formula that helps calculate the future value of an investment or loan where interest is continuously compounded. Here's how each component plays a role:

• P: Principal amount is the initial sum of money upon which interest grows.
• e^{rt}: Represents the exponential growth factor which changes depending on the rate of interest and time span.
• A: Is the future amount, reflecting the total accumulated principal and interest.

To effectively use these formulas, it's crucial to understand how to manipulate variables within the algebraic equations. For our exercise, we converted a percentage to a decimal, calculated the exponent, and then applied it to find the final investment outcome. This systematic approach is vital in working efficiently with algebraic formulas in various financial computations.