text: # A bacteria culture starts with 1,500 bacteria and doubles in size every 2 hours. How do you find an exponential model for the size of the culture as a function of time t in hours?
  
  Dec 17, 2016
  
  $s \left(t\right) = 1500 \cdot {2}^{\frac{t}{2}}$
  
  #### Explanation:
  
  If the culture has a growth rate, $r$ such that it doubles in size every $\textcolor{red}{2}$ hours, then
  $\textcolor{w h i t e}{\text{XXX}} {r}^{\textcolor{red}{2}} = 2$
  $\rightarrow$
  $\textcolor{w h i t e}{\text{XXX}} r = \sqrt{2}$
  
  and, given an initial size of $1 , 500$ bacteria, its size after $t$ hours would be:
  $\textcolor{w h i t e}{\text{XXX}} s \left(t\right) = 1500 \cdot {\left(\sqrt{2}\right)}^{t}$ or $1500 \cdot {2}^{\frac{t}{2}}$