Let x(t) ∈ [0, 1] be the fraction of maximum capacity of alive-music venue at time t (in hours) after the door opens. Therate at which people go into the venue is modeled by dx dt = h(x)(1− x), (1) where h(x) is a function of x only. 1. Consider the casein which people with a ticket but outside the venue go into it at aconstant rate h = 1/2 and thus dx dt = 1 2 (1 − x). (a) Find thegeneral solution x(t). (b) The initial crowd waiting at the doorfor the venue to open is k ∈ [0, 1] of the maximum capacity (i.e.x(0) = k). How full is the venue at t?
2. Suppose people also decides whether to go into the venuedepending on if the place looks popular. This corresponds to h(x) =3 2 x and thus dx dt = 3 2 x(1 − x). (a) Find the general solutionx(t). (b) What should be the initial crowd x(0) if the band wantsto start playing at t = 2 hours with 80% capacity?
3. Consider the two models, A and B, both starting at 10% fullcapacity. Model A is governed by the process of question (1) andmodel B is governed by the process described in question (2). Startthis question by writing down the respective particular solutionsxA(t) and xB(t). (a) Which of the two models will first reach 50%of full capacity? (b) Which of the two models will first reach 99%of full capacity? (c) Plot the curves xA(t) and xB(t). Both curvesshould be consistent with: (i) your answers to the two previousitems; (ii) the rate of change at t = 0 (i.e., dx dt at t = 0);(iii) the values of x in the limit t → ∞.
How do you do question 3?
