Answer to Question #315538 in Differential Equations for Allen

Question

Bacteria grow in a nutrient solution at a rate proportional to the amount present. Initially, there are 250 strands of the bacteria in the solution which grows to 800 strands after seven hours. Find (a) an expression for the approximate number of strands in the culture at any time t and (b) the time needed for the bacteria to grow to 1600 strands. 

Answer 1

Solution: Given a starting population of $N_0$, our general expression for exponential growth is $N=N_0e^{rt}$

where $r$ is a positive constant. Substituting for the given information, we can find $r$ as follows:

$800=250e^{r(7)}$

$3.2=e^{7r}$

$r=\frac{1}{7}ln(3.2) \\ \Rightarrow ~r=0.166164$

a) A general expression for the number of stands in the culture at any time is $N=250e^{0.166164t}$

b) The time needed for the bacteria to grow to 1,600 strands is therefore:

$1600=250e^{0.166164t} \\6.4=e^{0.166164t} \\t=\frac{1}{0.166164}ln(6.4) \\ \Rightarrow t=11.1715$

Therefore, it takes $11.1715\approx11.2$ hours for the population to grow to 1,600 strands.