EXPONENTLAL AND LOGARITHMIC FUNCTIONS Evaluating an exponential function with base e that models a... number of bacteria $P(t)$ in a certain population increases according to the following function, where time $t$ is measured in hours. \[ P(t)=2000 e^{0.16 t} \] nd the initial number of bacteria in the population and the number of bacteria after 7 hours. und your answers to the nearest whole number as necessary. Initial number: bacteria Number after 7 hours: bacteria
Solution
Step 1 ：The problem provides the function \(P(t)=2000 e^{0.16 t}\) which models the number of bacteria in a population over time, where \(t\) is the time in hours.
Step 2 ：To find the initial number of bacteria, we need to evaluate the function at \(t=0\). Substituting \(t=0\) into the function gives us \(P(0)=2000 e^{0.16 \times 0} = 2000\).
Step 3 ：So, the initial number of bacteria is \(\boxed{2000}\).
Step 4 ：To find the number of bacteria after 7 hours, we need to evaluate the function at \(t=7\). Substituting \(t=7\) into the function gives us \(P(7)=2000 e^{0.16 \times 7} \approx 6129.7084065860045\).
Step 5 ：Rounding this to the nearest whole number, we get approximately 6130.
Step 6 ：So, the number of bacteria after 7 hours is approximately \(\boxed{6130}\).
