- Identify the principal \(P = 1500\), rate \(r = 0.026\), frequency \(n = 365\), and time \(t = 3.5\).
- Apply the compound interest formula: \(A = P(1 + \frac{r}{n})^{nt}\).
- Calculate the final amount: \(A = 1500(1 + \frac{0.026}{365})^{1277.5} \approx 1642.898\).
- Round the final result to the nearest cent: \(\boxed{\$1642.90}\).
Explanation
- Problem Analysis and Variable Identification
The objective is to calculate the future value of a savings account given a principal amount, an annual interest rate, and a compounding schedule. We are provided with the following parameters:
- Principal amount (\(P\)) = \(\$1500\)
- Annual interest rate (\(r\)) = \(2.6\% = 0.026\)
- Compounding frequency (\(n\)) = \(365\) times per year (daily)
- Time period (\(t\)) = \(3.5\) years
We use the standard compound interest formula, which is given by:
\[A = P \left(1 + \frac{r}{n}\right)^{nt}\]
where \(A\) represents the final amount in the account.
- Calculation of the Final Amount
To find the final balance, we substitute the known values into the compound interest formula:
\[A = 1500 \left(1 + \frac{0.026}{365}\right)^{365 \times 3.5}\]
First, calculate the periodic interest rate:
\[\frac{0.026}{365} \approx 0.0000712328767\]
Then, determine the total number of compounding periods:
\[365 \times 3.5 = 1277.5\]
Now, compute the exponentiation:
\[A = 1500 \times (1.0000712328767)^{1277.5}\]
Calculating this expression results in:
\[A \approx 1500 \times 1.09526545556 \approx 1642.898183...\]
- Rounding and Conclusion
The calculated amount is approximately \(\$1642.898183\). Since the question asks for the amount rounded to the nearest cent, we look at the third decimal place. Because the third digit is \(8\), we round the second decimal place up from \(9\) to the next integer, effectively increasing the tens of cents. Therefore, \(\$1642.898...\) rounds to \(\$1642.90\).
The amount of money in the account after \(3.5\) years is \(\boxed{\$1642.90}\).
Examples
Compound interest is a fundamental concept in finance used to determine how savings or investments grow over time when interest is reinvested. For instance, if you deposit \(\$1500\) into a savings account with a \(2.6\%\) annual interest rate compounded daily, the interest earned each day is added to your balance, causing your money to grow at an increasing rate each subsequent day. Understanding this formula helps individuals make informed decisions about long-term savings strategies, such as how much an initial investment will be worth after several years.