Question

The population of a certain community is known to increase at a rate proportional to the number of people present any time.if the population had double in 5 years,how long will it take to triple?

Answer 1

Let $n$ be the number of people present any time. Then $y'=ky,$ and hence $\frac{dy}{y}=kdt.$ It follows that $\int\frac{dy}{y}=\int kdt,$ and thus $\ln|y|=kt+C,$ that is $y(t)=Ce^{kt}.$ It follows that $y(5)=2y(0),$ that is $Ce^{5k}=2C.$ We conclude that $k=\frac{1}{5}\ln 2,$ and consequently, $y(t)=Ce^{\frac{1}{5}(\ln 2)t}=C2^{\frac{t}{5}}.$ Let us find when the population will triple. It follows that $y(t)=3y(0),$ that is $C2^{\frac{t}{5}}=3C.$ We have that $t=5\log_2 3\approx 7.9$

Answer: 7.9 years