# Using the concept of derivatives solve the following linear model.  Using the concept of derivatives solve the following linear model. 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 triple? What will be the population in  years?

## Expert Answers The specific numbers did not appear in the problem. The general method is:

Given `(dp)/(dt)=kp` where `k` is a proportionality constant and `p` is the population at time `t` . We know that any function whose rate of change is proportional to the function itself has the following property:

`(dy)/(dx)=ky=> y=Ce^(kt)` with k the proportionality constant and C a constant.

Thus we have `p=Ce^(kt)` as the population function. You will take the two data points (the population at two different times) to solve for the two unknowns C and k using substitution.

Then with that function you can evaluate at any other time t.

In order to find when the population triples, you will solve for t given a known p.

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