Assignment M331Scaling Problems1. The equations X′t= −k1Xt, X0= AY′t= −k2Yt+ k1Xt, Y0= 0model the movement of a drug from the digestive system to the bloodstream. The modelinvolves three parameters, k1,k2and A. Scale the problem to reduce the model to just oneparameter.2. The equation M′t= P −MtVS, M0= 0models the amount of pollution in a lake of volume V when the pollution rate is given by Pand the rate of outflow from the lake is S. This equation contains three parameters, S,V,and P. Scale the problem to reduce the model to just one parameter.3. The equationsa ∂tCx,t= D∂xxCx,tx > 0,t > 0Cx,0= C0e−βxx > 0D∂xC0,t= H∂tC0,t, t > 0are used to model the movement of methane gas in soil. The model contains fourparameters, a, D, β and H. Scale the problem to reduce the model to just two parameters.4. The equationsLC ∂ttux,t+LG + RC∂tu + RG u = ∂xxux,tx > 0,t > 0ux,0= 0 x > 0u0,t= U0cosωt, t > 0are used to model the transmission of electrical impulses in a conductor. The modelcontains six parameters, L,G,R,C,ω.and U0Scale the problem to reduce the model to justtwo parameters.Unconstrained OptimizationUsing 620 feet of fence wire, build a rectangular pen of maximum area by making use ofthe fixed walls of length 300 feet and 400 feet, respectively as shown in the following sketch1Note that there are 3 different configurations that make use of the fixed walls.In each of these configurations, the area is equal to xjyjbut in each case, xjand yjsatisfya different condition (e.g. in the second configuration the condition is, x2+ y2= 620).(a) In each of the three configurations:express y in terms of x;what are the max and min values for x and y in each configuration?express the area in terms of x for 0 ≤ x ≤ xmax(b) determine the maximum area that can be achieved and discuss the natureof the extreme point where the max area occurs.(c) plot the graph of Ax versus x and show where the max occurs.(d) does your answer change if the vertical wall is 400 while the horizontal wall is