# Using P(t)=P_oe^(kt) how do you find the count in the bacteria culture was 900 after 20 minutes and 1100 after 35 minutes and find what was the inital size of the culture and the population after 95 minutes?

Jan 19, 2017

Initial size of the culture was $688$ and the population after $95$ minutes is $2455$.

#### Explanation:

As the count in the bacteria culture was $900$ after $20$ minutes and $P \left(t\right) = {P}_{o} {e}^{k t}$

we have $P \left(20\right) = 900 = {P}_{o} {e}^{20 t}$

Similarly as the count in the bacteria culture was $1100$ after $35$ minutes

we have $P \left(35\right) = 1100 = {P}_{o} {e}^{35 t}$

Dividing latter by former, we get $\frac{1100}{900} = \frac{11}{9} = {e}^{35 t - 20 t}$

i.e. ${e}^{15 t} = \frac{11}{9}$ and $15 t = \ln \left(\frac{11}{9}\right) = 0.013378$

and hence ${P}_{o} = \frac{900}{e} ^ \left(0.013378 \times 20\right) = \frac{900}{1.306772} = 688.72$

and the population after 95 minutes is $688.72 \times {e}^{95 \times 0.013378}$

= $688.72 \times {e}^{1.27091} = 2455$

**all calculations done using scientific calculator