The number of bacteria present in a Petri dish can be modeled by the function $N=50 e^{3 t}$, where $N$ is the number of bacteria present in the Petri dish after $t$ hours. Using this model, determine, to the nearest hundredth, the number of hours it will take for $N$ to reach 30,700 .
Solution
Step 1 :Solve the equation \(30,700 = 50 e^{3t}\) for \(t\)
Step 2 :Calculate \(t\) using the equation \(t = \frac{\ln(\frac{30,700}{50})}{3}\)
Step 3 :Simplify the final answer: \(t \approx 2.14\)