(2 points) Another model for a growth function for a limited pupulation is given by the Gompertz function, which is a solution of the differential equation dPdt=cln(KP)P where c is a constant and K is the carrying capacity. (a) Solve this differential equation for c=0.05, K=1000, and initial population P0=100. P(t)= . Hint: Note that dPdt=cln(KP)P=−c(lnP−lnK)P and then use Separation of Variables with u-substitution in the integration step (or do this mentally). (b) Compute the limiting value of the size of the population. limt→[infinity]P(t)= . (c) At what value of P does P grow fastest? P= . Hint: Differentiate dPdt again with respect to t (implicitly) and use the first derivative test to find the value of P that gives a maximum in dPdt.