Questions: Solve for (t) to two decimal places. [ 2=e^0.09 t ] (t approx) (Do not round until the final answer. Then round to the nearest hundredth as needed.)

Solution Steps

To solve the equation \(2 = e^{0.09t}\) for \(t\), we need to isolate \(t\). This can be done by taking the natural logarithm of both sides of the equation. The natural logarithm will allow us to bring down the exponent, making it possible to solve for \(t\).

We start with the equation given in the problem: \[ 2 = e^{0.09t} \]

To isolate \(t\), we take the natural logarithm of both sides: \[ \ln(2) = \ln(e^{0.09t}) \] Using the property of logarithms that \(\ln(e^x) = x\), we simplify the right side: \[ \ln(2) = 0.09t \]

Now, we can solve for \(t\) by dividing both sides by \(0.09\): \[ t = \frac{\ln(2)}{0.09} \] Calculating this gives us: \[ t \approx 7.7016 \] Rounding to two decimal places, we find: \[ t \approx 7.70 \]

Final Answer