Question

$500 were deposited into an account with a 8.5% interest rate, compounded continuously. how many years was it in the bank if the current amount is $3500? t = ? years

Explanation:

Step1: Write the continuous - compounding formula

The formula for continuous compounding is \(A = Pe^{rt}\), where \(A\) is the final amount, \(P\) is the principal amount, \(r\) is the interest rate (in decimal form), and \(t\) is the time in years.
Given \(P=\$500\), \(r = 0.085\), and \(A=\$3500\). Substitute these values into the formula: \(3500=500e^{0.085t}\).

Step2: Simplify the equation

Divide both sides of the equation by \(500\): \(\frac{3500}{500}=e^{0.085t}\), so \(7 = e^{0.085t}\).

Step3: Take the natural logarithm of both sides

Using the property \(\ln(e^{x})=x\), if \(7 = e^{0.085t}\), then \(\ln(7)=\ln(e^{0.085t})\). So \(\ln(7)=0.085t\).

Step4: Solve for \(t\)

We know that \(\ln(7)\approx1.94591\). Then \(t=\frac{\ln(7)}{0.085}\). Substitute \(\ln(7)\approx1.94591\) into the formula: \(t=\frac{1.94591}{0.085}\approx22.9\).

Answer:

\(t\approx22.9\) years