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.