Chapter 8.4, Problem 61PE

The population $P\left(t\right)$ of a culture of bacteria grows exponentially for the first $72\text{ hr}$ according to the model $P\left(t\right)=P_0{e}^{kt}.$ The variable $t$ is the time in hours since the culture is started. The population of bacteria is $60,000$ after $7\text{ hr}\text{.}$ The population grows to $80,000$ after $12\text{ hr}\text{.}$

a. Determine the constant $k$ to $3$ decimal places.

b. Determine the original population $P_0$ . Round to the nearest thousand.

c. Determine the time required for the population to reach $300,000$ . Round to the nearest hour.