- Use the continuous compounding formula \(A(t) = P e^{rt}\) with \(P = 6000\) and \(r = 0.045\).
- Calculate the balance after 3 years: \(A(3) = 6000 e^{0.045 \times 3} \approx \boxed{\$6,867.22}\).
- Solve for the time \(t\) to reach \(\$10,000\): \(10000 = 6000 e^{0.045t}\) leads to \(t = \frac{\ln(5/3)}{0.045} \approx \boxed{11.35 \text{ yr}}\).
Explanation
- Problem Analysis and Model Setup
We are given an initial principal investment of \(P = \$6,000\) that grows at an annual interest rate of \(r = 4.5\% = 0.045\), compounded continuously. The formula for the amount \(A\) at time \(t\) (in years) under continuous compounding is given by \(A(t) = P e^{rt}\). We need to determine the amount after \(3\) years and the time required for the account balance to reach \(\$10,000\).
- Calculating the Amount After 3 Years
To find the amount after \(3\) years, we substitute \(P = 6000\), \(r = 0.045\), and \(t = 3\) into the continuous compounding formula:
\[A(3) = 6000 e^{0.045 \times 3}\]
Calculating the exponent: \(0.045 \times 3 = 0.135\).
Evaluating the expression: \(A(3) = 6000 e^{0.135} \approx 6000 \times 1.144537 \approx 6867.2207\).
Rounding to the nearest cent, the amount after \(3\) years is \(\$6,867.22\).
- Calculating the Time to Reach $10,000
To determine how long it will take for the balance to grow to \(\$10,000\), we set \(A(t) = 10000\) and solve for \(t\):
\[10000 = 6000 e^{0.045t}\]
First, divide both sides by \(6000\):
\[\frac{10000}{6000} = e^{0.045t} \implies \frac{5}{3} = e^{0.045t}\]
Next, take the natural logarithm (\(\ln\)) of both sides:
\[\ln\left(\frac{5}{3}\right) = 0.045t\]
Finally, solve for \(t\):
\[t = \frac{\ln(5/3)}{0.045} \approx \frac{0.510826}{0.045} \approx 11.35168\]
Rounding to two decimal places, it will take \(11.35\) years for the investment to grow to \(\$10,000\).
- Final Answer
The amount after \(3\) years is \(\$6,867.22\), and the time required for the account balance to reach \(\$10,000\) is \(11.35\) years.
Examples
Continuous compounding is a fundamental concept in finance, often used to model the growth of long-term investments or retirement savings. For instance, if you decide to set aside a portion of your income into a savings account that compounds continuously, understanding these formulas allows you to forecast your financial future. Whether you are planning for a down payment on a home or retirement, knowing how your money grows over time helps you make informed decisions about your savings goals and time horizons.