Problem 30 - Precalculus Mathematics for Calculus

Question

Compound Interest An investment of \(\$ 7,000\) is deposited into an account in which interest is compounded continuously. Complete the table by filling in the amounts to which the investment grows at the indicated times or interest rates. \(t=10\) years $$\begin{array}{|c|c|} \hline \begin{array}{c} \text { Rate } \\ \text { per year } \end{array} & \text { Amount } \\ \hline 1 \% & \\ 2 \% & \\ 3 \% & \\ 4 \% & \\ 5 \% & \\ 6 \% & \\ \hline \end{array}$$

Step-by-Step Solution

Answer

Use the continuous compounding formula \( A = 7000 \times e^{rt} \) with each interest rate to compute the amounts.

Step 1: Understand the formula for continuous compounding

The formula for calculating the future value of an investment compounded continuously is given by \( A = Pe^{rt} \), where \( A \) is the amount of money accumulated after time \( t \), \( P \) is the principal amount (\$7,000 in this case), \( r \) is the annual interest rate, and \( t \) is the time in years.

Step 2: Calculate the amount for 1% interest

Substitute the values into the formula for \( r = 0.01 \) and \( t = 10 \) years.\[ A = 7000 \times e^{0.01 \times 10} \] Calculate \( A \).

Step 3: Calculate the amount for 2% interest

Substitute the values into the formula for \( r = 0.02 \) and \( t = 10 \) years.\[ A = 7000 \times e^{0.02 \times 10} \] Calculate \( A \).

Step 4: Calculate the amount for 3% interest

Substitute the values into the formula for \( r = 0.03 \) and \( t = 10 \) years.\[ A = 7000 \times e^{0.03 \times 10} \] Calculate \( A \).

Step 5: Calculate the amount for 4% interest

Substitute the values into the formula for \( r = 0.04 \) and \( t = 10 \) years.\[ A = 7000 \times e^{0.04 \times 10} \] Calculate \( A \).

Step 6: Calculate the amount for 5% interest

Substitute the values into the formula for \( r = 0.05 \) and \( t = 10 \) years.\[ A = 7000 \times e^{0.05 \times 10} \] Calculate \( A \).

Step 7: Calculate the amount for 6% interest

Substitute the values into the formula for \( r = 0.06 \) and \( t = 10 \) years.\[ A = 7000 \times e^{0.06 \times 10} \] Calculate \( A \).

Key Concepts

Continuous Compounding

Continuous compounding is a powerful concept in finance, where interest is added to the principal balance of an investment or loan at every possible instant. Unlike annual, quarterly, or monthly compounding, continuous compounding assumes that the reinvestment of interest occurs an infinite number of times in a year.

This method uses the natural exponential function represented as \( e^{rt} \) to calculate the future value. Here, \( e \) is a constant approximately equal to 2.71828, and it plays a crucial part in modeling growth processes.

Continuous compounding is best visualized as a constant growth curve, where the investment grows smoothly without interruption. This concept is particularly significant for long-term investments where reinvestment of interest can lead to substantial growth.

Exponential Growth

Exponential growth describes a scenario in finance where the value of an investment increases rapidly over time, influenced by a constant rate of growth. In the context of continuous compounding, this growth is depicted by the formula \( A = Pe^{rt} \).

Here, the amount \( A \) becomes larger at a rate proportional to its current size, leading to faster accumulation over time. This characteristic of exponential growth ensures that even small increases in the interest rate can significantly impact the future value of an investment.

For example, a principal invested at a higher interest rate compounded continuously will grow considerably more over the same period than one with a lower rate. Understanding exponential growth helps investors appreciate how quickly wealth can build when left undisturbed.

Interest Rate Calculations

Calculating interest rates accurately is key to maximizing investment returns. With continuous compounding, the annual interest rate \( r \) directly affects the exponential growth of an investment. To compute the future value, the rate is multiplied by the time period to find the exponent of \( e \) in the formula.

When dealing with interest rates:

• Convert percentage rates to decimals; for instance, 1% becomes 0.01.
• Ensure the time period \( t \) is in years for consistency.
• Analyze how changes in \( r \) impact the overall growth.

By carefully inputting these values into the continuous compounding formula, investors can project the exact growth of their initial principal. It allows them to make well-informed decisions regarding which rates would best serve their financial goals.

Future Value Formula

The future value formula with continuous compounding, expressed as \( A = Pe^{rt} \), enables investors to calculate the amount an investment will grow to at a specific future point in time.

In this formula:

• \( A \) represents the future value of the investment.
• \( P \) is the principal amount—the initial sum of money invested.
• \( r \) is the annual interest rate (expressed as a decimal).
• \( t \) is the time the money is invested for, in years.

The formula's exponential nature allows for smooth growth of the investment over time, illustrating how effectively wealth can be accumulated with continuous compounding. This methodology provides clarity on the potential returns of various financial strategies based on different interest rates and time frames.