Question

an initial investment of $100 is now valued at $150. the annual interest rate is 5%, compounded continuously. the equation $100e^{0.05t} = 150$ represents the situation, where $t$ is the number of years the money has been invested. about how long has the money been invested? use your calculator and round to the nearest whole number. \boxed{} years

Explanation:

Step1: Isolate the exponential term

Divide both sides of the equation \(100e^{0.05t}=150\) by 100.
\(\frac{100e^{0.05t}}{100}=\frac{150}{100}\)
Simplifies to \(e^{0.05t}=1.5\)

Step2: Take the natural logarithm of both sides

Take \(\ln\) of both sides: \(\ln(e^{0.05t})=\ln(1.5)\)
Using the property \(\ln(e^x)=x\), we get \(0.05t = \ln(1.5)\)

Step3: Solve for \(t\)

Divide both sides by 0.05: \(t=\frac{\ln(1.5)}{0.05}\)
Calculate \(\ln(1.5)\approx0.4055\), then \(t=\frac{0.4055}{0.05}\approx8.11\)

Step4: Round to the nearest whole number

Rounding 8.11 to the nearest whole number gives 8.

Answer:

8