Question

Logarithm 5. Wilma spits into a petri dish and, using a microscope, determines that there are approximately4000 bacteria cells. The bacteria cells grow continuously according to the model P= Poe^kt where P is the population of bacterial cells after + hours. After 12 hours, it is determined that there are approximately 12500 cells in the dish. What is the growth rate of the cells?

Answer 1

Given data:

`\begin{gathered} P_0=4000 \\ t=12 \\ P=12500 \\ \\ P=P_0e^(kt) \end{gathered}`

Use the given data in the model and solve k (growth rate):

`12500=4000e^(12k)`

Divide both sides of the equation into 4000:

`\begin{gathered} (12500)/(4000)=(4000)/(4000)e^(12k) \\ \\ (25)/(8)=e^(12k) \end{gathered}`

Find the natural logarithm of both sides of the equation:

`\begin{gathered} \ln ((25)/(8))=\ln (e^(12k)) \\ \\ \ln ((25)/(8))=12k \end{gathered}`

Divide both sides of the equation by 12:

`\begin{gathered} (\ln ((25)/(8)))/(12)=(12)/(12)k \\ \\ (\ln((25)/(8)))/(12)=k \\ \\ \\ k=(\ln((25)/(8)))/(12) \end{gathered}`

Evaluate:

`k\approx0.095`

Then, the growth rate is 0.095 (9.5%)