# A bacteria culture starts with 1000 bacteria. 2 hours later there are 1500 bacteria. How do you fin an exponential model for the size of the culture as function of time t in hours, and use the model to predict how many bacteria there will be after 2 days?

Oct 27, 2016

$P = 16834112$

#### Explanation:

So and exponential growth or decay will fit the formula,

$P = {P}_{\text{0}} {e}^{r \cdot t}$

where

$P = \text{current population at time t}$
${P}_{\text{0"= "starting population}}$
$r = \text{rate of exponential growth/decay}$
$t = \text{time after start}$

so we sub in what we know

$1500 = 1000 {e}^{r \cdot 2}$

$\frac{3}{2} = {e}^{r \cdot 2}$

$\ln \left(\frac{3}{2}\right) = r \cdot 2$

$r = \frac{\ln \left(3\right) - \ln \left(2\right)}{2}$

$r = 0.20733$

so our population is modelled by,

$P = 1000 \cdot {e}^{\left(0.20733\right) \cdot t}$ , t hours after start

so 2 days is 48 hours so subbing in,

$P = 1000 \cdot {e}^{\left(0.20733\right) \cdot \left(48\right)}$

$P = 16834112$