Gymnast clothing manufactures expensive soccer cleats for sale to college bookstores in runs of up to 500. its cost (in dollars) for a run of x pairs of cleats is c(x) = 3100 + 9x + 0.1x2 (0 ≤ x ≤ 500). gymnast clothing sells the cleats at $100 per pair. find the revenue and profit functions. how many should gymnast clothing manufacture to make a profit?

  Revenue: R(x) = 100x
  Profit: P(x)= 3100 + 9x + 0.1x2     (0 ≤ x ≤ 500)

equalize the two equations
100x=3100 + 9x + 0.1x2

0.1x2-91x+3100=0
using graphic tools
x1=35.447----35
x2=874 
the domain is (0 ≤ x ≤ 500)

the value from which there are profits is 35 pairs of cleats

Revenue: R(x) = 100x --------100*35=$3500
Profit: P(x)= 3100 + 9x + 0.1x2 -------3100+9*35+0.1*(35)2=$3537.5
Profit > Revenue  up to the amount of 35  pairs of cleats