Population \(y\) grows according to the equation \(\dfrac{dy}{dt}=ky\), where \(k\) is a constant and \(t\) is measured in years. If the population doubles every 5 years, then the value of \(k\) is

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Seperate variables and integrate:

$$ \frac{dy}{dt}=ky $$ $$ \frac{1}{y}  dy=k   dt $$ $$ \ln|y|=kt+C $$ $$ y=Ae^{kt} $$

At \(t=0\), the population is \(A\).

$$ y|_{t=0} = Ae^{k(0)} $$ $$ y=Ae^0 $$ $$ y=A $$

At \(t=5\), the population should double to \(2A\).

$$ 2A=Ae^{k(5)} $$ $$ 2=e^{5k} $$ $$ \ln2=5k $$ $$ \frac{\ln2}{5}=k $$ $$ k \approx \boxed{0.139} $$

The amount of time required for a quantity to double (assuming continous growth) is:

$$ \text{doubling time} = \frac{\ln 2}{\text{growth rate}} $$

If we are given the doubling time, we can use it to solve for the growth rate, which is what the constant \(k\) refers to in our population growth model.