Question

3
multiple choice 1 point
derrick invests $5000 in a savings account that earns 6% annual interest, compounded continuously. how long will it take to double his money?
about 5 years
about 12 years
about 15 years
about 50 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 annual interest rate (in decimal form), and \(t\) is the time in years.
Given \(P=\$5000\), \(r = 0.06\), and \(A = 2P=\$10000\). Substitute these values into the formula: \(10000=5000e^{0.06t}\).

Step2: Simplify the equation

Divide both sides of the equation \(10000 = 5000e^{0.06t}\) by \(5000\). We get \(2=e^{0.06t}\).

Step3: Take the natural logarithm of both sides

Using the property \(\ln(e^{x})=x\), take the natural logarithm of both sides of \(2 = e^{0.06t}\). So, \(\ln(2)=\ln(e^{0.06t})\), which simplifies to \(\ln(2)=0.06t\).

Step4: Solve for \(t\)

We know that \(\ln(2)\approx0.693\). Then \(t=\frac{\ln(2)}{0.06}\). Substitute \(\ln(2)\approx0.693\) into the formula: \(t=\frac{0.693}{0.06}=11.55\approx12\).

Answer:

about 12 years