Question

A bacteria culture grows with constant relative growth rate. After 2 hours there are 400 bacteria and after 8 hours the count is 50,000. Find the rate of growth after 7 hours. P'(7) = ?

Answer 1

i was thinking to take the derivative of C*exp(r*7)

Answer 2

the initial population of P(0) = 80

Answer 3

C = 80

Answer 4

i tried taking the natural log of 80*exp(r*7) and the taking log differentials

Answer 5

\[\Large P(t)=C \cdot e^{r \cdot t}\]

\[\large 400=Ce^{2r} \]\[\large 50000=Ce^{8r} \]

\[\large\implies {500\cancel{00}\over 4\cancel{00}}={\cancel Ce^{8r}\over \cancel Ce^{2r}}\]\[\implies 125=e^{6r}\\\implies6r=\ln(125)=\ln \left( 5^3 \right)=3\ln(5)\\\implies \color{blue}{r={\ln \left( 5 \right)\over2}} \approx 0.804718956\]

plugging this r in the first equation ... we can confirm \(C=80\)