Problem 60 - College Algebra with Corequisite Support

Question

The formula for the amount \(A\) in an investment account with a nominal interest rate \(r\) at any time \(t\) is given by \(A(t)=a(e)^{r t},\) where \(a\) is the amount of principal initially deposited into an account that compounds continuously. Prove that the percentage of interest earned to principal at any time \(t\) can be calculated with the formula \(I(t)=e^{r t}-1\)

Step-by-Step Solution

Answer

The percentage of interest earned is \( e^{r t} - 1 \).

Step 1: Understand the Problem

We need to prove that the percentage of interest earned on the principal can be calculated by \( I(t) = e^{r t} - 1 \) given that the total amount at time \( t \) is \( A(t) = a(e)^{r t} \), where \( a \) is the initial principal.

Step 2: Express Interest Earned Formula

The interest earned at any time \( t \) is defined as the difference between the total amount \( A(t) \) and the initial principal \( a \). Therefore, the interest earned can be expressed as \( I(t) = A(t) - a \).

Step 3: Substitute for A(t)

Substitute the formula \( A(t) = a(e)^{r t} \) into the expression for interest. This gives us: \( I(t) = a(e)^{r t} - a \).

Step 4: Factor Out the Principal

Factor the initial principal \( a \) out of the expression: \( I(t) = a((e)^{r t} - 1) \).

Step 5: Percentage Interest to Principal Ratio

To find the percentage of interest to the principal, divide the interest earned by the principal \( a \): \( \frac{I(t)}{a} = \frac{a((e)^{r t} - 1)}{a} = (e)^{r t} - 1 \).

Step 6: Conclusion

The percentage of interest earned to principal at any time \( t \) is given by \( I(t) = (e)^{r t} - 1 \), as required.

Key Concepts

Compound Interest

Compound interest is a powerful concept in finance and investing. It means that the interest earned over time not only applies to the initial principal but also on the interest that accumulates in each period.
When interest is "compounded," the amount grows at a faster rate than with simple interest where the interest is only calculated on the principal.
This exponential growth can significantly increase your investment earnings over longer periods. Understanding the impact of compounding can be crucial for your investments:

• With each interval, the interest is added to the principal, forming a new base for the next calculation.
• The frequency of compounding affects growth: annual, semi-annual, quarterly, monthly, or even daily.
• More frequent compounding periods will lead to more accumulated interest over time.

By applying compound interest, what feels like modest growth each period can turn into significant financial gains over the long term.

Continuous Compounding

Continuous compounding is a unique and theoretical form of compound interest, where the frequency of compounding is infinite.
This means that interest is compounded constantly, leading to a slightly higher amount than discrete compounding methods.
It's often represented by the formula, \[A(t) = a imes e^{r imes t}\]where \(A(t)\) is the amount, \(a\) is the principal, and \(e\) is the constant approximately equal to 2.71828.Continuous compounding has a few interesting characteristics:

• It represents the maximum growth an investment can achieve under a given interest rate.
• Since it's mathematically continuous, it finds practical applications in more sophisticated financial models.
• Although not often used in real-world bank accounts, it helps illustrate the potential growth limit of an investment.

Understanding this concept is beneficial for maximizing investment strategies and realizing the full potential of compounded returns in theoretical models.

Interest Rate Formula

The interest rate formula is central to calculating the growth of investments over time.
It ties together the principal amount, the rate of interest, and the period of time to determine the final amount or the interest earned.
In continuous compounding, the formula \[I(t) = e^{r imes t} - 1\]calculates the percentage of interest earned relative to the principal.Important points about the interest rate formula include:

• The exponent \(r imes t\) determines how powerfully the amount grows, linking the rate and time.
• The term \(e^{r \times t}\) reflects the effect of compounding over time.
• Subtracting 1 isolates the interest earned portion from the total compounded amount.

This formula particularly emphasizes the role of time and rate in growing investments, demonstrating how strategic financial planning can amplify returns. Better understanding these nuances enables more effective financial decision-making.