Problem 38 - 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. $$ P=33,999, r=-4 \%, t=21 \text { years } $$

Step-by-Step Solution

Answer

\(A \approx 14,680.93\)

Step 1: Understand the Formula

The given formula is used to calculate the amount \(A\) after applying continuous compounding interest to a principal \(P\). The formula is \(A = Pe^{rt}\), where \(r\) is the rate of interest and \(t\) is the time.

Step 2: Convert the Rate to Decimal

The given interest rate \(r\) is \(-4\%\). To use it in the formula, convert it to a decimal by dividing by 100: \(r = \frac{-4}{100} = -0.04\).

Step 3: Substitute Values into the Formula

Now, substitute the values into the formula: \(P = 33,999\), \(r = -0.04\), and \(t = 21\). The equation becomes \(A = 33,999 \cdot e^{-0.04 \times 21}\).

Step 4: Calculate the Product of \(r\) and \(t\)

Calculate \(-0.04 \times 21\), which gives \(-0.84\). This will be used as the exponent for \(e\).

Step 5: Find the value of \(e^{-0.84}\)

Using a calculator, compute \(e^{-0.84}\). The value is approximately 0.4317.

Step 6: Compute \(A\)

Multiply the principal \(P\) by the value of \(e^{rt}\): \(A = 33,999 \times 0.4317\).

Step 7: Round the Final Result

Calculate the product: \(A \approx 33,999 \times 0.4317 = 14,680.93\). Round this to the nearest hundredth to get \(14,680.93\).

Key Concepts

Exponential Functions

Exponential functions are a fundamental part of mathematics and are extensively used in finance, biology, and many other fields. They involve equations where a constant base is raised to the power of a variable. In the context of finance, these functions help in modeling situations where things grow or decay at a constant proportional rate.
For example, the exponential function in continuous compounding interest is written as:

• \( A = Pe^{rt} \).

Here, \( e \) is the base of the natural logarithm, approximately equal to 2.71828. This function captures the essence of continuous growth or decay, where rather than computing interest periodically, it is compounded at every possible moment.
Exponential functions model real-life phenomena such as population growth, radioactive decay, and in this scenario, financial growth or decline because of compounding interest. Mastery of these concepts allows you to predict future values efficiently.

Interest Rate Conversion

Interest rates are often expressed in percentages, so converting them into a decimal form is crucial for calculations. This process is necessary when working with formulas involving exponential functions.
In our example, the interest rate of 4% needs to be expressed in decimal for computational purposes. This is done by dividing by 100:

• \( r = \frac{-4}{100} = -0.04 \)

Converting interest rates ensures the correct input into the exponential formula so that the final computation reflects accurate financial scenarios, such as gains or losses over time.

Mathematical Calculations

Mathematical calculations are fundamental to finance for determining values such as future investments. To calculate accurately, breaking each step into manageable parts is beneficial.
First, substitute all known values into the appropriate formula — in this case, \( A = Pe^{rt} \). Calculate the exponent by multiplying \( r \) (the converted rate) and \( t \) (time in years):

• \( -0.04 \times 21 = -0.84 \)

After determining the exponent, calculate \( e^{-0.84} \) using a calculator, which results in approximately 0.4317.
Finally, multiply the principal amount by this value:

• \( A = 33,999 \times 0.4317 \approx 14,680.93 \)

At every step, check for precision, especially when using devices for calculations. Rounding the final result to the nearest hundredth ensures a practical approach in terms of real-world money usage.