# Equilibrium Solutions

1. Sep 15, 2010

### lionsgirl12

1. Problem:
In order for a movie to become a box office hit, its popularity (as in positive audience's words-of-mouth, and buzz in the press) must have risen past a certain critical threshold T among viewers and potential viewers, before and during the first week of its nation-wide release. Else it quickly fades away, relegated to second-run theaters ("two-thumbs down..."). If the film's popularity should surpass the threshold level, it will become a bona fide hit. Therefore, after production and marketing expenses, it would return a tidy profit for the production company. However, if its popularity should ever climb further higher and beyond a break-out level, B, the movie would become a true blockbuster and its ticket sale would skyrocket - sometimes to an enormous level. Until a saturation level, K, when the market's spending capacity is reached. As the marketing director for the motion picture High School Musical IV: College Musical, you expect, based on extensive marketing researches, that the film's popularity can be modeled by the following autonomous equation (in million of ticket-buying children spending parents' money):

p' = -p(2p-12)(3p-48)2(p-43)​

(a) Identify the values of T, B, and K, and classify the stability of their respective equilibrium solution.

(b) If p(1066) = 6, to what value will p approach eventually?

(c) If p(2010) = 16.0000025, to what value will p approach eventually?

2. Relevant equations:
equilibrium sollutions occur whenever y' = f(y) = 0

3. The attempt at a solution:
T = critical threshold
B = break-out level
K = saturation level

p' = p(2p-12)(3p-48)2(p-43)
-p=0
p=0

2p-12=0
p=6

(3p-48)2=0
p=16

p-43=0
p=43