1. The population of a town grows at a rate proportional to the population present at time t. The initialpopulation of 500 increases by 5% in 10 years.a. What will be the population in 60 years? (Round your answer to the nearest person.)b. How fast (in persons/yr) is the population growing at t = 60? (Round your answer to two decimalplaces.)

A differential equation can be solved using the variable separation method. In this method, the terms with like variables are isolated on either side of the equation and then integrated with respect to the variables to yield the general solution. The general solution of a differential equation always contains a constant of integration. The initial condition in an initial value problem is used to determine the integration constant.Let the initial population be P at anytime t. The growth rate is dPdt, and it is proportional to the present population, that is, P. Hence, the differential equation can be formulated as dPdt=kP, where k is the constant of proportionality. One initial condition is provided as P(0)=500. The population increases by 5% in 10 years. Hence, the population in 10 years is 500+500×5100=525. Therefore, the other initial condition is P(10)=525.Solve the differential equation dPdt=kP using the variable separation method. dPdt=kPdPP=kdt∫dPP=k∫dtln P=kt+C Here, C is the integration constant.Use the initial condition P(0)=500 to calculate the integration constant C. ln P=kt+Cln 500=k(0)+C=C Substitute C=ln 500 in the differential equation ln P=kt+C and simplify to obtain the differential equation as follows: ln P=kt+C=kt+ln 500ln P-ln 500=ktlnP500=ktP500=ektP=500ekt Hence, the resulting equation is P=500ekt.Use the other initial condition P(10)=525 to calculate k. P=500ekt525=500ek(10)e10k=525500ln e10k=ln 1.0510k=ln 1.05k=ln 1.0510 Substitute k=ln 1.0510 in the equation P=500ekt and simplify it to obtain the particular solution. Use eln x=x for simplification. P=500ekt=500eln 1.0510t=500eln 1.05t10=5001.05t10 Hence, the solution is P=5001.05t10, where P is the population at time t (measured in years).Substitute t=60 in the equation P=5001.05t10 to calculate the population in 60 years. P=5001.05t10=5001.056010=5001.056≈670 Hence, the population is about 670 in 60 years.The growth rate is dPdt, and it is defined by the equation dPdt=kP. Substitute k=ln 1.0510 and P=670 in the equation dPdt=kP to calculate the growth for t=60. dPdt=kP=ln 1.0510×670=ln 1.05×67≈3.27 Hence, the growth rate is about 3.27 persons per year.