Determine the growth constant, $$ k $$, of a population that is growing at a rate proportional to its size, where the population triples every 11 days.

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Given the exponential growth model

$$
P(t)=P_0e^{kt},
$$

and knowing that the population triples every 11 days (i.e.,

$$
P(11)=3P_0),
$$

we substitute into the model:

$$
3P_0=P_0e^{11k}.
$$

Dividing both sides by $$ P_0 $$ gives:

$$
3=e^{11k}.
$$

Taking the natural logarithm of both sides results in:

$$
\ln 3 = 11k.
$$

Solving for $$ k $$ yields:

$$
k=\frac{\ln 3}{11}.
$$
The growth constant $$ k $$ is $$ \frac{\ln 3}{11} $$.

Determine the growth constant, , of a population that is growing at a rate proportional to its size, where the population triples every 11 days.

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