# The count in a bacteria culture was 100 after 20 minutes and 2000 after 40 minutes. Assuming the...

## Question:

The count in a bacteria culture was 100 after 20 minutes and 2000 after 40 minutes. Assuming the count grows exponentially, what was the initial size of the culture?

## Exponential Growth:

All exponential behaviors have a growth constant that describes the proportionality that exists in the function. An exponential function is used to describe situations that have data that changes a lot in each period of time.

Data:

First Time: {eq}t=20\:\textrm{minutes} {/eq}

First Future Value: {eq}F=100 {/eq}

Second Time: {eq}t=40\:\textrm{minutes} {/eq}

Second Future Value: {eq}F=2000 {/eq}

Equation:

{eq}F=Pe^{kt} {/eq}

Then,

{eq}\begin{align*} F&=Pe^{kt} \\ \\ 100&=Pe^{20k} \\ \\ P&=\frac{100}{e^{20k}} \\ \\ 2000&=Pe^{40k} \\ \\ P&=\frac{2000}{e^{40k}} \\ \\ \frac{2000}{e^{40k}}&=\frac{100}{e^{20k}} \\ \\ \frac{2000}{100}&=\frac{e^{40k}}{e^{20k}} \\ \\ 20&=e^{40k-20k} \\ \\ 20&=e^{20k} \\ \\ e^{20k}&=20 \\ \\ 20k&=\ln \left(20\right) \\ \\ k&=\frac{\ln \left(20\right)}{20} \\ \\ k&=0.15 \end{align*} {/eq}

Now, the initial size of the culture will be:

{eq}\begin{align*} P&=\frac{100}{e^{20k}} \\ \\ P&=\frac{100}{e^{20 \cdot 0.15}} \\ \\ P&=\boxed{5} \end{align*} {/eq} 