Problem 3 - Calculus and its applications

Question

Find the future value \(P\) of each amount \(P_{0}\) invested for time period t at interest rate \(k\), compounded continuously. $$ P_{0}=\$ 140,000, \quad t=9 \mathrm{yr}, \quad k=5.8 \% $$

Step-by-Step Solution

Answer

The future value is approximately $235,900.

Step 1: Understand the Formula

To find the future value of an investment compounded continuously, we use the formula: \( P = P_{0} e^{kt} \) , where \( P \) is the future value, \( P_{0} \) is the initial amount, \( k \) is the interest rate as a decimal, and \( t \) is the time in years.

Step 2: Convert the Interest Rate

The interest rate \( k \) is given as 5.8%. To use it in the formula, convert it to a decimal by dividing by 100. Thus, \( k = \frac{5.8}{100} = 0.058 \).

Step 3: Substitute Values into the Formula

Substitute \( P_{0} = 140,000 \), \( k = 0.058 \), and \( t = 9 \) into the formula \( P = P_{0} e^{kt} \). This gives us the equation \( P = 140,000 \times e^{0.058 \times 9} \).

Step 4: Calculate Exponent and Future Value

Calculate the exponent \( 0.058 \times 9 = 0.522 \). Then calculate \( e^{0.522} \) using a calculator, which is approximately \( 1.685 \). Multiply this by the initial amount: \( P = 140,000 \times 1.685 \approx 235,900 \).

Step 5: Conclusion and Round the Result

The future value of the investment, rounded to the nearest dollar, is approximately \( 235,900 \).

Key Concepts

Interest Rate Conversion

Before we can dive into calculations, it's essential to work with the right form of the interest rate. When an interest rate is given as a percentage, it can't be directly utilized in mathematical formulas without conversion. This is why we need to convert percentage rates into decimal form. To convert, simply divide the interest rate by 100. For example, if you have 5.8%, convert it to a decimal by calculating \(\frac{5.8}{100} = 0.058\).

• This conversion standardizes the rate for equations.
• It improves accuracy in calculations.
• This step is crucial for correct future value computations.

Whether you’re using simple interest, compound interest, or continuous compounding, starting with the correct form of the interest rate sets the foundation for accurate financial computations.

Future Value Calculation

Calculating the future value of an investment predicts how much it will grow over a specified period under certain conditions. Continuous compounding is a unique interest scenario where your investment grows at an infinite number of time intervals. This continuous growth can be calculated using the formula \( P = P_{0} e^{kt} \).

• \( P \) is the future value.
• \( P_{0} \) is the initial investment or principal.
• \( e \) is the base of the natural logarithm, approximately equal to 2.71828.
• \( k \) is the annual interest rate in decimal form.
• \( t \) represents time in years.

By calculating the exponent \( kt \) and then finding \( e^{kt} \), you can determine how much your investment will be worth at the end of your investment period. This method stands out because it assumes compounding occurs at every possible instant.

Exponential Growth Formula

Continuous compounding exemplifies exponential growth because it maximizes the potential for an investment's value increase. Exponential growth in investments reflects how powerful interest calculations are over time, particularly using the natural base \( e \).
To illustrate, the formula \( P = P_{0} e^{kt} \) encompasses each component of exponential growth:

• The principal \( P_{0} \) is your investment's current baseline.
• The expression \( e^{kt} \) captures how your investment grows exponentially, not linearly.

As \( kt \) increases, so does \( e^{kt} \), meaning that even small changes in either rate or time can significantly magnify the investment's future value.
When compounding continuously, the effects of exponential growth are maximized, emphasizing the importance of understanding compounding's impact to capitalize on investment opportunities.