A bacteria culture starts with 500 bacteria and doubles in size every 6 hours. How do you find an exponential model for t?

Sep 25, 2017

Exponential model of bacteria growth at time $t$ is
${B}_{t} = 500 \cdot {e}^{0.115525 t}$

Explanation:

Starting number of bacteria is ${B}_{0} = 500$ , It doubles in

size every $6$ hours. Number of bacteria after $t = 6$ hours

grows to ${B}_{6} = {B}_{0} \cdot 2 = 500 \cdot 2 = 1000$. The exponential growth

formula is ${B}_{t} = {B}_{0} \cdot {e}^{k t}$ , where $k$ is growth rate

$\therefore {e}^{k t} = {B}_{t} / {B}_{0} \mathmr{and} {e}^{k t} = \frac{1000}{500} = 2$ Taking natural

log on both sides we get $\ln {e}^{k t} = \ln 2 \mathmr{and} k t = \ln 2$ (since

$\ln {e}^{k t} = k t$) $\therefore k = \ln \frac{2}{t} = \ln \frac{2}{6} \mathmr{and} k \approx 0.115525$

Exponential model of bacteria growth at time $t$ is

${B}_{t} = 500 \cdot {e}^{0.115525 t}$ [Ans]