The population of a town grows at a rate proportional to the population present at time t. the initial population of 50 increases by 15% in 10 years. what will be the population in 30 years? how fast is the population growing at t=30?

I have a solution here however there is a slight change in the given. The initial population is 500 instead of 50. By analyzing this solution, this could help you solve the problem on your own!

So the first part: "The population of a town grows @ a rate porportional to the pop. present @ time t." Means. 

dP = A*P. Where P is the population at the time 't' and A is some constant (thus the 'proportional' part). 

Then we've got: "the initial pop. of 500 increases by 15% in 10 years". Which means: 

P(0) = 500 (This says Population at time 0 years, or initial population, equals 500) and P(10) = 500*1.15 = 575 (Population at time 10 years). 

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