Answer:
The initial population is 6598.
Step-by-step explanation:
Given : The population of a community is known to increase at a rate proportional to the number of people present at time t. The initial population [tex]p_o[/tex] has doubled in 5 years. Suppose it is known that the population is 10,000 after 3 years.
To find : What was the initial population [tex]p_o[/tex] ?
Solution :
Using the formula, [tex]P(t)=P_oe^{kt}[/tex]
The initial population [tex]p_o[/tex] has doubled in 5 years.
i.e. [tex]P(5)=2P_o[/tex]
For t=5, [tex]P(5)=P_oe^{5k}[/tex]
[tex]2P_o=P_oe^{5k}[/tex]
[tex]2=e^{5k}[/tex]
[tex]\log 2=5k[/tex]
[tex]k=\frac{\log 2}{5}[/tex]
Substitute in the equation,
[tex]P(t)=P_oe^{(\frac{\log 2}{5})t}[/tex]
[tex]P(t)=P_o2^{\frac{t}{5}}[/tex]
Substitute, P(t)=10,000 and t=3 years
[tex]10000=P_o2^{\frac{3}{5}}[/tex]
[tex]P_o=\frac{10000}{2^{\frac{3}{5}}}[/tex]
[tex]P_o=6598[/tex]
Therefore, The initial population is 6598.
