Problem 60 - College Algebra

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 any time \(t\) can be calculated with the formula \(I(t)=e^{r t}-1\)

Step-by-Step Solution

Answer

The interest percentage is given by \(I(t) = e^{rt} - 1\).

Step 1: Understand the Problem

We need to prove that the percentage of interest earned compared to the principal in a continuously compounding investment account over time \(t\) follows the formula \(I(t) = e^{rt} - 1\).

Step 2: Define the Relationship Between Amount and Principal

The formula for the amount \(A(t)\) in an account is given as \(A(t) = a \cdot e^{rt}\). Here, \(a\) represents the principal, \(r\) the interest rate, and \(t\) the time. This represents the value of the investment after time \(t\).

Step 3: Express Interest Amount Earned

The interest earned at any time \(t\) is the amount in the account minus the principal, i.e., \(I(t) = A(t) - a\). Substituting for \(A(t)\), we get \(I(t) = a \cdot e^{rt} - a\).

Step 4: Factor Out the Principal

Factor the principal \(a\) out of the equation: \(I(t) = a(e^{rt} - 1)\). This formula gives the absolute value of interest earned, factoring out the initial principal.

Step 5: Calculate Percentage of Interest

To find the percentage of interest earned relative to the principal, divide the interest amount by the principal: \[ \text{Percentage Interest} = \frac{I(t)}{a} = \frac{a(e^{rt} - 1)}{a} = e^{rt} - 1 \]

Step 6: Conclusion

Therefore, the percentage of interest earned as compared to the principal over time \(t\) is indeed \(I(t) = e^{rt} - 1\), proving the formula.

Key Concepts

Exponential Growth

Exponential growth is a fascinating phenomenon where quantities increase at a rate proportional to their current value.
This creates a situation where the growth becomes faster and faster over time.
In the context of continuously compounded interest, the growth of an investment is represented by an exponential function, displayed as:

• \( A(t) = a \cdot e^{rt} \)

Here, \(e\) is the base of the natural logarithm, approximately equal to 2.71828.
The "\(e^{rt}\)" part of the formula captures the exponential nature of growth with respect to the interest rate \(r\) and time \(t\).
As the value of \(t\) increases, the amount \(A(t)\) grows exponentially faster compared to simple or linear growth models.
This means small increases in time or interest rate can lead to substantial growth in investment, illustrating the powerful effect of exponential growth.

Interest Rate

The interest rate, represented by \(r\) in the formula, is a crucial factor impacting how quickly your investment grows.
It shows how much growth or return you can expect from the principal amount over a specific time period, usually a year.
A higher interest rate implies quicker and larger returns on an investment, especially with continuously compounding interest.

• The formula used, \( A(t) = a \cdot e^{rt} \), depicts the role of \(r\) as a multiplier of time \(t\).

This interaction gives the exponential factor \(e^{rt}\) its power, as both "\(r\)" and "\(t\)" increase the potential return.
Notably, in continuous compounding, the interest is added instantaneously, leading to constant growth influenced by even the smallest rate change.
Compared to ordinary compounding (daily, monthly, or annually), continuous compounding maximizes the benefits of the given interest rate.

Principal Amount

The principal amount, denoted as \(a\), is the initial sum of money you invest or deposit.
It's the foundation of the continuously compounding interest formula:

• \( A(t) = a \cdot e^{rt} \)

The principal serves as the base amount that earns interest.
Over time, the principal grows as it earns interest continuously at the rate \(r\), magnifying the total amount through the power of exponential growth.
Factoring out the principal also helps in calculating the exact interest earned, as shown by:

• \( I(t) = a(e^{rt} - 1) \)

This equation isolates the growth accrued by the principal alone, providing a clear picture of how much interest has been earned relative to the initial amount.
Thus, the principal is key as it determines the scale of the investment's growth potential.