Question

The population P, t years after the initial observation is given by the formula:

p_t=(100000)/(2+3e^-0.05t)

Determine the exact size of the population and the time in years when the rate of growth is maximum

Answer 1

$$P(t)=\dfrac{100000}{2+3e^{-0.05t}}$$

$$P'(t)=\dfrac{15000e^{-0.05t}}{(2+3e^{-0.05t})^2}=\dfrac{15000e^{0.05t}}{(3+2e^{0.05t})^2}$$

$$P''(t)=\dfrac{750e^{0.05t}(3-2e^{0.05t})}{(3+2e^{0.05t})^3}$$

$$P''(t)=0=>\dfrac{750e^{0.05t}(3-2e^{0.05t})}{(3+2e^{0.05t})^3}=0$$

$$e^{0.05t}=1.5$$

$$t=20\ln 1.5$$

The rate of growth is maximum at $t=20\ln 1.5.$

Then

$$P(20\ln 1.5)=\dfrac{100000}{2+3(2/3)}=25000$$

The exact size of the population is 25000 and the time is $20\ln 1.5\approx8.11$ years when the rate of growth is maximum.