Problem 1: Consider a time-varying population that follows the logistic population model. The population has a limiting population of 800 , and at time t=0, its population of 200 is growing at a rate of 60 per year. a) Write a differential equation for the population P(t) that models this scenario. Solution: dtdP=rP(1KP) known: P= population at t r= growth/yr, 60 k= limiting population, 800 \%plug known into original formula dtdP=60P(1800P) b) Identify an appropriate solution technique and solve this differential equation. Solution: \%divide both sides by P"(1-P/800) \%multiply both sides by dt \% multiply both sides by ot P(1800P)dP=60dt \%integrate c) Use MATLAB to plot your solution for t=0 to t=10. Solution: \$scode d) How long will it take for the population to achieve 95% of its maximum population? Solution: Put your math/explanation here... e) Use the MATLAB dsolve() function to solve the differential equation and plot the solution on a new figure. The results provided by dsolve() and your solution from part (b) should match