Problem 77 - Elementary and Intermediate Algebra

Question

assume that there are no deposits or withdrawals. Determining the Initial Deposit. \(\quad\) An account now contains \(\$ 11,180\) and has been accumulating interest at \(7 \%\) annual interest, compounded continuously, for 7 years. Find the initial deposit.

Step-by-Step Solution

Answer

The initial deposit is approximately $7,000.

Step 1: Understand the Problem

We analyze the given problem to determine what is being asked.

Step 2: Set Up the Solution

The formula for continuously compounded interest is given by: \[ A = Pe^{rt} \]where:- \( A \) is the amount of money accumulated after n years, including interest.- \( P \) is the principal amount (the initial deposit).- \( r \) is the annual interest rate (in decimal).- \( t \) is the time the money is invested for in years.- \( e \) is the base of the natural logarithm.We need to find the initial deposit \( P \).

Step 3: Solve the Problem

Applying the relevant mathematical techniques, we work through the solution step by step.

Step 4: State the Result

The initial deposit is approximately $7,000.

Key Concepts

Interest Rate

The interest rate is a crucial part of understanding how investments grow over time, particularly with continuously compounded interest. When we talk about the interest rate, we are referring to the percentage at which your money grows each year. This rate is usually presented as a yearly percentage and signifies how much extra you earn based on your initial deposit.

For continuously compounded interest, this interest rate has a unique role. It is used in a special formula that helps calculate how much your money will be worth in the future. The formula integrates the interest rate as a key factor, using it to determine the exponential growth of your investment.

In our example, the interest rate is 7%, which means that annually, your initial deposit grows at a rate of 7%. It's important to note that this is expressed as a decimal in calculations, so 7% becomes 0.07. This conversion is critical for plugging into the continuous compound interest formula correctly.

Initial Deposit

The initial deposit, also known as the principal, is the amount of money initially invested or deposited into an account before any interest is added. Understanding the initial deposit is important because it forms the foundation of your investment or savings.

When calculating the future value of an investment using continuously compounded interest, the initial deposit represents the starting point from which your money grows. In the context of this problem, we are tasked with finding this initial deposit amount given the future value of $$11,180.

• The formula used for continuous compounding is: \[ A = Pe^{rt} \]
• Here, \( A \) is the accumulated amount, and you need to solve for \( P \) (initial deposit) when \( A = 11,180 \).

This initial deposit calculation allows us to understand how much money we initially put in that would grow to a certain amount, given a specific interest rate and time period. The initial deposit provides the baseline for all subsequent growth generated by the interest rate over the specified timeline.

Natural Logarithm

The natural logarithm is a powerful mathematical concept that plays a significant role in solving problems related to continuously compounded interest. In particular, it helps us find the initial deposit when we have the accumulated amount after interest has been applied.

The natural logarithm, often denoted as \( \ln \), is the logarithm to the base \( e \), where \( e \) is a mathematical constant approximately equal to 2.71828. This concept is crucial for breaking down the exponential growth seen in continuously compounded interest.

To find the initial deposit in our scenario, we rearrange the formula:

• \[ A = Pe^{rt} \]
• to solve for \( P \), resulting in \[ P = \frac{A}{e^{rt}} \]

You would typically use the natural logarithm to isolate variables when calculating backwards from the total accumulated amount back to the initial deposit.
By taking the natural logarithm, we can manipulate the equation to make \( P \) the subject, allowing easy calculation when you know the interest rate and time period. This makes solving and understanding continuous compounding much more accessible.