Question

the rate of change of a population P is proportional to P. if P=100 when t= 0 and P=800 when t=3, what is P(6)?

Answer 1

Given the population growth formula as shown below

`P=P_0e^(rt)`
`\begin{gathered} r\rightarrow rate \\ P_0\rightarrow population\text{ at time t=0} \\ t\rightarrow time \\ \end{gathered}`

When time is equal to zero

`\begin{gathered} t=0 \\ P_0=100 \\ P=100 \end{gathered}`
`\begin{gathered} when \\ t=3 \\ 800=100e^(r(3)) \\ 800=100e^(3r) \\ (800)/(100)=e^(3r) \end{gathered}`
`\begin{gathered} e^(3r)=8 \\ lne^(3r)=ln8 \\ 3r=2.0794 \\ r=(2.0794)/(3)=0.6931 \end{gathered}`

Thus

`\begin{gathered} t=6 \\ P=100e^(0.6931*6) \\ P=6398.19 \end{gathered}`

Hence, the value of P(6) is approximately 6398.19