Problem 33 - 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. $$ P=5,000, r=8 \%, t=20 \text { years } $$

Step-by-Step Solution

Answer

\( A \approx 24765.00 \)

Step 1: Convert the Interest Rate

Convert the percentage interest rate into a decimal by dividing by 100. Given \( r = 8\% \), the decimal form is \( r = \frac{8}{100} = 0.08 \).

Step 2: Substitute Given Values Into Formula

Substitute the given values of \( P = 5000 \), \( r = 0.08 \), and \( t = 20 \) into the compound interest formula: \[ A = 5000 \cdot e^{0.08 \times 20} \]

Step 3: Calculate the Exponent

Calculate the exponent by multiplying the interest rate by the time: \[ 0.08 \times 20 = 1.6 \]

Step 4: Calculate the Exponential Function

Find the value of \( e^{1.6} \) using a calculator. \[ e^{1.6} \approx 4.953 \]

Step 5: Calculate the Final Amount

Multiply the principal by the exponential value to find \( A \): \[ A = 5000 \times 4.953 = 24765 \]

Step 6: Round the Solution

Round \( 24765 \) to the nearest hundredth. Since it is a whole number, it remains \( 24765.00 \).

Key Concepts

Exponential Growth

Exponential growth represents a process where a quantity increases rapidly over time by a constant proportion at each time interval. In finance, exponential growth is often seen in situations involving compound interest. Here, the interest grows on both the initial principal and the accumulated interest from previous periods, causing the amount to increase significantly over time.
To visualize this, consider a snowball rolling down a hill. Initially, it begins small, but it quickly grows larger as more snow accumulates. This is similar to how your investment can grow under exponential growth. When money is invested under exponential growth conditions, its value over time can be determined using the formula:

• \( A = P e^{rt} \)

where:

: the principal amount (initial investment)
e: the base of the natural logarithm (approx. 2.718)
: the rate of growth (as a decimal)
: the time period the money is invested for
This powerful formula showcases how the combination of principal, rate, and time results in exponential growth.

Interest Rate Conversion

When dealing with interest rates, it is common to see them expressed as percentages. However, for calculations involving compound interest, these percentages must be converted into a decimal form. This step is crucial, as it allows for the proper integration of the interest rate within mathematical formulas, such as the compound interest formula we used.
Converting percentages to decimals is straightforward: divide the percentage by 100.

• For example, an 8% interest rate becomes 0.08 in decimal form.
• This ensures that the formula \( A = Pe^{rt} \) can be applied correctly.

Failing to convert the interest rate properly can lead to incorrect calculations and a misunderstanding of the actual growth potential of an investment. Always ensure your rates are in the correct decimal form before proceeding with further calculations.

Continuous Compounding

Continuous compounding refers to the scenario where an investment's interest is calculated and reinvested into the principal continuously rather than at discrete intervals such as annually, quarterly, or monthly. This method results in the highest possible return because the investment experiences continuous exponential growth.
The formula for continuous compounding is represented as:

• \( A = Pe^{rt} \)

Here, e is the base of the natural logarithm, approximately 2.718, representing the continuous growth factor. The term \(e^{rt}\) signifies the continuous application of the interest rate over time.
Continuous compounding maximizes the growth of an investment by accumulating interest instantly and constantly. This makes the method more powerful than discrete compounding intervals. In practical applications, knowing how to calculate returns using continuous compounding is beneficial for understanding the full potential of an investment.