Problem:
The supply function of a product is given by p(x) = 400 + 0.8x - 0.2*SQRT(x).
Determine the producer surplus when the sales level is 121.
Solution:
Surplus = INT [p(x)] dx from 0 to 121
= INT [400 + 0.8x - 0.2*SQRT(x)] dx from 0 to 121
= INT [400 + 0.8*x^1 - 0.2*x^(1/2)] dx from 0 to 121
= [400*x + (0.8/2)x^2 - 0.2 /(3/2)*x^3/2] from 0 to 121
= [400*x + 0.4*x^2 - 1/5 /(3/2)*x^3/2] from 0 to 121
= [400*x + 0.4*x^2 - 2/15*x^3/2] from 0 to 121
= [400*121 + 0.4*121^2 - 2/15*121^3/2] - [400*0 + 0.4*0^2 - 2/15*0^3/2]
= [48,400 + 5,856.4 - 177.47] - [0 + 0 - 0]
= 48,400 + 5,856.4 - 177.47
= 54,078.93
THEREFORE, the Producer surplus is 54,079 products
