Problem 67 - Precalculus : Building Concepts and Connections

Question

For an initial deposit of \(\$ 1500,\) find the total amount in a bank account after \(t\) years for the interest rates and values of t given, assuming continuous compounding of interest. \(3.25 \%\) interest; \(t=5.5\)

Step-by-Step Solution

Answer

The total amount in the bank account after 5.5 years is approximately \$1896.45.

Step 1: Identifying the known variables

Identify the given values: initial deposit \(P = \$1500\), annual interest rate \(r = 3.25\% = 0.0325\) when converted to decimal, and time \(t = 5.5\) years.

Step 2: Applying the formula

Substitute the known values into the formula for continuous compound interest, \( A = Pe^{rt} \).

Step 3: Calculating the total amount

After substituting the known values, the formula becomes \( A = 1500 * e^{(0.0325 * 5.5)} \). Computing this will provide the total amount in the bank account after 5.5 years.

Key Concepts

Initial Deposit

When opening a bank account, the initial amount of money that you put in is called the initial deposit. In our exercise, the initial deposit was \(\$1500\). This is the starting point from which your savings will grow over time due to interest.An initial deposit is important because it's the base amount that will accumulate interest. The larger your initial deposit, the more interest you can earn over time. It's like planting a seed that you want to grow into a tree; the bigger the seed, the bigger the tree can potentially be once matured.

Interest Rate Calculation

The interest rate is a key factor in determining how fast your money grows in a bank account. In this exercise, the interest rate was provided as \(3.25\%\). For calculations, it is important to convert this percentage into a decimal, which is done by dividing by 100. Therefore, \(3.25\% = 0.0325\).Banks offer different types of interest rates:

• Simple Interest: Interest is calculated only on the initial deposit.
• Compound Interest: Interest is calculated on the initial deposit plus any accumulated interest.

In our case, we are using continuous compounding. This means interest is calculated and added to the account continuously, resulting in more frequent compounding than typical intervals, like annually or monthly.

Exponential Growth

Continuous compound interest results in exponential growth. This means as time goes on, the amount of interest earned grows at an increasing rate. In our exercise, we used the formula for continuous compound interest, which is given by:\[A = Pe^{rt}\]- \(A\) is the amount of money accumulated after \(t\) years, including interest.- \(P\) is the initial deposit.- \(e\) is the base of the natural logarithms, approximately equal to 2.718.- \(r\) is the annual interest rate (expressed as a decimal).- \(t\) is the time the money is invested for in years.The magic of exponential growth with continuous compounding is that the more time your money is invested, the more significant the increase. The interest you earn is constantly added to the base amount, allowing it to grow faster as time goes on.