At a time denoted as  t = 0  a technological innovation is introduced into a community that has a fixed population of n people. Determine a differential equation for the number of people x(t) who have adopted the innovation at time t if it is assumed that the rate at which the innovations spread through the community is jointly proportional to the number of people who have adopted it and the number of people who have not adopted it. (Use  k > 0  for the constant of proportionality and x for  x(t).  Assume that initially one person adopts the innovation.)

One method of slowing the growth of an insect population without using pesticides is to introduce into the population a number of sterile males that mate with fertile females but produce no offspring. Let P represent the number of female insects in a population and S the number of sterile males introduced each generation. Let r be the per capita rate of production of females by females, provided that their chosen mate is not sterile. Then the female population is related to time t by t = ∫(P + S)dP/P[(r-1)P - S]   Suppose an insect population with 10,000 females grows at a rate of r = 1.2 and 900 sterile males are added initially. Evaluate the integral to give an equation relating the female population to time. (Note that the resulting equation cannot be solved explicitly for P.)

On a monthly basis, when a factory employs L worker-hours of labor, they can produce Q(L) = 7L + 40L/6 scooters. An investment in labor of K dollars gives them an employment level of L(K)=0.055K – 12,500 K worker-hours. The factory currently invests 2,000 dollars in labor. At what rate is the number of scooters being produced changing with respect to the amount they invest in labor? Round your final answer to 2 decimal places and include units with your answer. а.