# Ans this question The population of a certain community is known to increase at a rate proportional to the number of people present at any time. The population grows from people to people in...

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The population of a certain community is known to increase at a rate proportional to the number of people present at any time. The population grows from people to people in years. How long will it take for the original population to double? What will be the population in years?

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### 1 Answer

The population grows at a rate that is proportional to the number of people at any time.

Use the notation P for population and t for time.

The equation expressing the proportionality relation between the rate of growth and the number of people at any time is : `(dP)/(dt) = n*P`

`` n denotes a constant of proportionality.

Rewrite this equation such that `(dP)/P = ndt.`

Integrating both sides yields`int (dP)/P = int ndt=gt ln P = n*t + c`

`P(t) = c*e^(nt)`

To determine the constant c, you need to know what is the number of people in a certain number of years.

If you need to know the time taken for the population to double, you need to know the initial number of people.

If you need to know the number of people in a certain number of years, you should replace t by the number of years, in equation `P(t) = c*e^(nt).`