Problem 41 - Algebra 2

Question

An investor withdraws all \(\$ 525\) from an account that was neglected for 8 years. It earned 3.4\(\%\) annual interest, compounded continuously. How much was the initial deposit?

Step-by-Step Solution

Answer

The initial deposit was approximately \$377.07

Step 1: Identify the values given in the problem

From the problem, the final amount \(A = \$525\), the interest rate \(r = 3.4\%\) or \(0.034\) when expressed as a decimal, and the time period \(t = 8\) years.

Step 2: Apply the values to the continuous compound interest formula

Substitute \(A = \$525\), \(r = 0.034\), and \(t = 8\) years to the formula of continuously compounded interest \(A = Pe^{rt}\). This results in the equation \( 525 = Pe^{0.034 * 8}\).This equation needs to be solved for \(P\).

Step 3: Solve for the initial deposit P

In order to isolate \(P\), divide both sides of the equation by \(e^{0.034 * 8}\), resulting in \(P = \frac{525}{e^{0.034 * 8}} \). Now calculate the exact value of the decimal using a calculator.

Key Concepts

Exponential Growth

Exponential growth occurs when the increase of a quantity is proportional to its current amount, leading to faster and faster growth as time goes on. In the context of compound interest, which involves money, this principle results in your initial investment growing at an increasing rate. With each passing period, the amount of interest added not only multiplies but also compounds, stacking upon itself, which accelerates the growth even further.

The mathematical representation of exponential growth in continuously compounded interest is expressed with the formula \(A = Pe^{rt}\), where:

• \(A\) is the final amount in the account (or accumulated value).
• \(P\) is the initial principal (or initial deposit).
• \(e\) is the base of the natural logarithm, approximately equal to 2.71828.
• \(r\) is the annual interest rate.
• \(t\) is the time in years.

The exponential term \(e^{rt}\) illustrates how quickly your investment grows over time when compounded continuously. Understanding how this formula works helps us appreciate why continuous compounding leads to more rapid growth compared to ordinary, simpler forms of interest accumulation.

Interest Rate

The interest rate is a crucial factor in determining how fast your money grows. It represents the percentage of the principal that is paid as interest over a specific period. In our specific example, the interest rate is given as 3.4%.

When dealing with compound interest, especially continuous compounding, it's important to convert the percentage into a decimal form before using it in calculations. This is done by dividing the percentage value by 100. For a 3.4% rate, you'd convert it to a decimal as follows:

• 3.4% = 0.034

The interest rate not only affects how much interest you'll earn, but it also plays a role in the exponential growth calculation. A higher interest rate results in greater accumulation over time. Thus understanding its impact is vital when anticipating how much an initial deposit will grow over time.

Initial Deposit Calculation

Calculating the initial deposit required to reach a certain amount after a period of time with continuous compounding involves some simple mathematical steps.

In our problem, we know the final amount \(A\), the interest rate \(r\), and the time \(t\), and we need to find the initial deposit \(P\). As per the continuous compounding formula, we rearrange it to solve for \(P\):

• \(A = Pe^{rt}\) becomes \(P = \frac{A}{e^{rt}}\).

Substitute the known values into the formula:

• \(A = 525\), \(r = 0.034\), and \(t = 8\).

Now, calculate \(e^{0.034 \times 8}\) using a scientific calculator and divide 525 by this calculated value to get the initial deposit \(P\). This problem-solving process demonstrates how rearranging formulas and understanding interest principles help unravel the growth mystery of your investments.