Problem 26 - Algebra 2

Question

Find the amount in a continuously compounded account for the given conditions. principal: \(\$ 950\) annual interest 6.5\(\%\) time: 10 yr

Step-by-Step Solution

Answer

The final amount in the continuously compounded account is determined by the calculation provided in the third step. The exact value would depend on the computational tool used, but it's around \$1789.94.

Step 1: Identify the parameters

Let's identify the parameters from the question. The principal amount \( P \) is \$950, the annual interest rate \( r \) is 6.5\(\%\) or 0.065 in decimal form, and the time \( t \) in years is 10.

Step 2: Substitute the values into the formula

Substitute the parameters identified in step 1 into the formula for continuous compound interest, which provides \( A = 950e^{(0.065)(10)} \).

Step 3: Calculate the expression

By performing the calculation provided in step 2, the final amount \(A\) is obtained. This can be done using a scientific calculator to calculate the exponent.

Key Concepts

Principal Amount

The principal amount is the initial sum of money that you start with in an investment or savings account. In our example, this is the starting amount of $950 that you deposit into a bank account. This initial investment will grow over time with interest.

For continuously compounded interest, the principal is a crucial piece of the puzzle because it's the base amount that will earn interest.

• In simple words, it's your seed money.
• The larger the principal, the more interest you can earn over time.

Understanding how the principal works will help you make strategic decisions about how much to invest initially. It is important to remember that the principal amount does not change by itself; it only grows because of the interest it earns.

Annual Interest Rate

The annual interest rate is a percentage that determines how much interest your principal will earn over the course of a year. It is an indicator of the investment's yield and can vary depending on economic conditions and banking policies.

For continuously compounded interest, the interest rate is denoted as a decimal. In our case, the rate of 6.5% is written as 0.065.

• A higher rate means more growth.
• The interest is compounded continuously, which involves calculating interest at an infinitely small fraction of time throughout the year.

This means your money grows at a rate that's exponential. By converting the percentage into a decimal format, it becomes easier to apply it in the formulas needed for calculations.

Exponential Growth

Exponential growth refers to the increasing rate at which the value of an account grows due to continuously compounded interest. Unlike simple interest, where you earn interest on your initial principal only, exponential growth involves earning interest on your interest.

In the context of our problem, we employ the formula for continuous compounding, which is given by:\[A = Pe^{rt}\]
In this equation:

• \(A\) represents the amount of money accumulated after time \(t\), including both the initial principal and the interest.
• \(P\) is the principal amount.
• \(r\) is the annual interest rate in decimal form.
• \(t\) stands for the time the money is invested or borrowed in years.

The factor \(e^{rt}\) shows how the growth of money follows an exponential trajectory. Simply put, each dollar in the account grows more over time because it's earning interest on itself continuously. Exponential growth highlights why continuous compound interest is so powerful.