Question

answer each question and submit your work in the space provided.
how many years does it take an investment to double at an annual interest rate of 9.2% compounded continuously? (10 points)

Explanation:

Step1: Use 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.

We want to find \(t\) when \(A = 2P\) and \(r=0.092\). Substitute \(A = 2P\) into the formula: \(2P=Pe^{0.092t}\).

Since \(P
eq0\) (if \(P = 0\), there is no investment), we can divide both sides of the equation by \(P\). We get \(2=e^{0.092t}\).

Step2: Take the natural logarithm of both sides

Using the property \(\ln(e^{x})=x\), if \(2 = e^{0.092t}\), then \(\ln(2)=\ln(e^{0.092t})\).

So, \(\ln(2)=0.092t\).

Step3: Solve for \(t\)

We know that \(\ln(2)\approx0.6931\). Then \(t=\frac{\ln(2)}{0.092}\).

Substitute \(\ln(2)\approx0.6931\) into the formula: \(t=\frac{0.6931}{0.092}\approx7.53\)

Answer:

It takes approximately \(7.53\) years for the investment to double.