- Identify the principal \(P = \$6,000\), annual interest rate \(r = 0.05\), and time \(t = 2.5\) years.
- Use the formula \(A = P(1+r)^t\) to set up the calculation \(A = 6000(1.05)^{2.5}\).
- Evaluate the compounding factor to find the final balance \(A \approx \$6,778.36\).
- Round the result to match the given choices: \(\boxed{\$6,778}\).
Explanation
- Problem Analysis and Variable Identification
We are given that Mario invests a principal amount \(P = \$6,000\) in an account with an annual interest rate of \(r = 5\% = 0.05\). The interest is compounded annually, and we are asked to find the account balance \(A\) after \(t = 2.5\) years using the provided formula \(A = P(1+r)^t\). Our objective is to substitute the given values into the formula and calculate the final amount.
- Calculation of Account Value
Substituting the known variables \(P = 6,000\), \(r = 0.05\), and \(t = 2.5\) into the formula \(A = P(1+r)^t\), we get:
\[A = 6,000(1 + 0.05)^{2.5}\]
\[A = 6,000(1.05)^{2.5}\]
To calculate the expression, we first compute the compounding factor \((1.05)^{2.5}\). Evaluating this value gives approximately \(1.129726\). Multiplying this factor by the principal amount of \(\$6,000\):
\[A \approx 6,000 \times 1.129726 = 6,778.3579...\]
Rounding to the nearest dollar, the value of the account is approximately \(\$6,778\).
- Conclusion
Comparing our result with the provided options, we see that \(\$6,778\) matches the fourth option. Therefore, the value of the account after 2.5 years is \(\boxed{\$6,778}\).
Examples
Calculating compound interest is a fundamental skill for financial planning. For instance, if you decide to open a high-yield savings account or invest in bonds, understanding how your money grows over time allows you to make informed decisions about your future finances. This formula helps you see the impact of interest rates and time on your savings, demonstrating the power of compounding as your money grows exponentially rather than linearly.