Question

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 P0 has doubled in 5 years. Suppose it is known that the population is 10,000 after 3 years. What was the initial population P0? (Round your answer to one decimal place.)

Answer 1

The initial population is 6598.

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
`p_o` 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
`p_o` ?

Solution :

Using the formula,
`P(t)=P_oe^(kt)`

The initial population
`p_o` has doubled in 5 years.

i.e.
`P(5)=2P_o`

For t=5,
`P(5)=P_oe^(5k)`

`2P_o=P_oe^(5k)`

`2=e^(5k)`

`\log 2=5k`

`k=(\log 2)/(5)`

Substitute in the equation,

`P(t)=P_oe^{((\log 2)/(5))t}`

`P(t)=P_o2^{(t)/(5)}`

Substitute, P(t)=10,000 and t=3 years

`10000=P_o2^{(3)/(5)}`

`P_o=\frac{10000}{2^{(3)/(5)}}`

`P_o=6598`

Therefore, The initial population is 6598.