A particular factory manufactures (n) units per month.Each unit sells for (c) dollars.The number of units sold varies with the selling price according to ;
n=600-10c
and the monthly operating cost of the factory is given by;
C=$4000+12n
The monthly profit;
P=nc-C is a maximum when the unit selling price is ???

Question

A particular factory manufactures (n) units per month.Each unit sells for (c) dollars.The number of units sold varies with the selling price according to ;
   n=600-10c
and the monthly operating cost of the factory is given by;
  C=$4000+12n
The monthly profit;
   P=nc-C  is a maximum when the unit selling price is ???

anonymous · Answer

first, substitute n into the C equation to get:
C=4000+12(600-10c)= 11200-120c

then substitute n and C into P equation;
P=c(600-12c)-(11200-120c)
  = -12(c^2)+720c-11200

differentiate P in term of c:
  dP/dc = -24c+720

note that the 2nd derivative is negative. therefore dP/dc is a max point.

now let dP/dc=0, u will get c.