Questions: The number of bacteria P(t) in a certain population increases according to the following function, where time t is measured in hours. P(t)=3000 e^(0.09 t) Find the initial number of bacteria in the population and the number of bacteria after 7 hours. Round your answers to the nearest whole number as necessary.

The number of bacteria P(t) in a certain population increases according to the following function, where time t is measured in hours.

P(t)=3000 e^(0.09 t)

Find the initial number of bacteria in the population and the number of bacteria after 7 hours.
Round your answers to the nearest whole number as necessary.

Transcript text: The number of bacteria $P(t)$ in a certain population increases according to the following function, where time $t$ is measured in hours. \[ P(t)=3000 e^{0.09 t} \] Find the initial number of bacteria in the population and the number of bacteria after 7 hours. Round your answers to the nearest whole number as necessary.

Solution

Solution Steps

Step 1: Find the Initial Number of Bacteria

The initial number of bacteria in the population, denoted as $P_0$, is given directly and equals 3000.

Step 2: Calculate the Number of Bacteria After Time $t$

To find the number of bacteria at any given time $t$, we use the formula: \[P(t) = P_0 e^{rt}\] Substituting the given values, $P_0 = 3000$, $r = 0.09$, and $t = 7$, we get: \[P(7) = 3000 e^{0.09(7)} = 5632.832\]