Question

Let x(t) ∈ [0,1] be the fraction of maximum capacity of a live-music venue at time t (in hours) after the door opens. The rate 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 case in which people with a ticket but outside the venue go into it at a constant rate h = 1/2 and thus dx/dt = 1/2 (1−x).

(a) Find the general solution x(t).

(b) The initial crowd waiting at the door for 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 venue depending on if the place looks popular. This corresponds to

h(x) = 3/2x and thus dx/dt = 3 2 x(1−x).

(a) Find the general solution x(t).

(b) What should be the initial crowd x(0) if the band wants to start playing at t = 2 hours with 80% capacity?

3. Consider the two models, A and B, both starting at 10% full capacity. Model A is governed by the process of question (1) and model B is governed by the process described in question (2). Start this question by writing down the respective particular solutions xA(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 curves should be consistent with: (i) your answers to the two previous items; (ii) the rate of change at t = 0 (i.e., dx dt at t = 0); (iii) the values of x in the limit t →∞.

find the 3rd question solution all parts

Answer 1

Note: See the software generated graphs of XA(t) (red) and XB(t) (blue). See that the red graph touches the green line(50% capacity) first and the blue graph touches the violet line(99% capacity) first.