Problem 35 - 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. $$ P=20,000, r=10.5 \%, t=50 \text { years } $$

Step-by-Step Solution

Answer

The value of \( A \) is approximately \( 3,811,326 \).

Step 1: Convert Rate to Decimal

The interest rate given is in percentage, so we need to convert it to a decimal. The rate \( r = 10.5\% \) should be written as a decimal by dividing by 100: \( r = \frac{10.5}{100} = 0.105 \).

Step 2: Substitute Values into the Formula

Now that we have the rate as a decimal, substitute the values of \( P, r, \) and \( t \) into the formula \( A = P e^{r t} \). This gives us: \( A = 20000 \, e^{0.105 \times 50} \).

Step 3: Calculate the Exponent

Compute the exponent \( r \times t \): \( 0.105 \times 50 = 5.25 \). So the expression becomes \( A = 20000 \, e^{5.25} \).

Step 4: Evaluate the Exponential Expression

Use a calculator to find the value of \( e^{5.25} \). This evaluates to approximately 190.5663.

Step 5: Compute Final Result

Multiply the result from Step 4 by \( P \): \( A = 20000 \times 190.5663 \). This gives \( A \approx 3,811,326 \).

Step 6: Round to the Nearest Hundredth

The value calculated needs to be rounded to the nearest hundredth. The result \( A = 3,811,326 \) is already a whole number, so it does not need further rounding.

Key Concepts

Compound Interest

Compound interest is a powerful concept where interest is calculated not just on the initial principal amount, but also on the accumulated interest from previous periods. It allows your investment to grow at an exponential rate over time. Unlike simple interest, which only grows linearly, compound interest builds on itself, accelerating growth.

Here's why it matters:

• **Reinvestment:** As your interest earns more interest, your investment or savings grow faster.
• **Compounding Frequency:** The more frequently interest is compounded, the more you earn. Different banks or investment plans might compound interests annually, semi-annually, quarterly, monthly, or even daily.
• **Time:** The longer you leave your investment to compound, the larger it will grow. Starting early can significantly impact the total value accumulated over time due to this exponential growth.

With compound interest, even a small investment can grow significantly large over time if left to grow under favorable conditions.

Percent to Decimal Conversion

Converting a percentage to a decimal is a basic, yet essential math skill, especially when dealing with interest rates in financial calculations. A percentage is simply a way to express a number as a fraction of 100.

To convert a percentage to a decimal, follow these steps:

• **Remove the Percent Sign:** Start by taking away the percent sign '%'.
• **Divide by 100:** Move the decimal point two places to the left, effectively dividing the percentage by 100. For example, 10.5% becomes 0.105.

This conversion is crucial because most mathematical formulas, especially those involving interest rates, require rates to be in decimal form. It ensures consistency in calculations and helps avoid large errors.

Exponential Functions

Exponential functions are a key mathematical concept, especially when dealing with growth processes such as compound interest. An exponential function is usually written in the form \( y = a \cdot b^{x} \), where:

• **a** is the initial amount or starting value.
• **b** is the base of the exponential, representing the growth factor.
• **x** is the exponent, often related to time or number of periods.

In finance, the natural exponential function \( e^{x} \) is particularly important. The formula \( A = P e^{rt} \) involves the constant \(e\) (approximately 2.71828), known as Euler's number, which is foundational for continuous growth calculations.

Exponential growth functions are used to model scenarios where growth rates are proportional to the value already existing, just like how your interest grows in compound interest settings. The fascinating aspect of exponential functions is how quickly they can escalate, emphasizing the importance of understanding these functions when planning long-term financial strategies.