A manufacturer estimates that when q thousand units of a particular commodity ar

Question

A manufacturer estimates that when q thousand units of a particular commodity are produced each month, the total cost will be C(q)=0.9q^2 + 2q + 810 thousand dollars, and all q units can be sold at a price of p(q) = 22.2 1.2q dollars per unit.

1. the average cost per unit, AC(q) = C(q)/ q , is minimized at a level of production equal to:

(a) 20 thousands (b) 30 thousands (c) 40 thousands (d) 50 thousands (e) The average cost per unit has no minimum

2. The minimal average cost is: (a) 27 dollars per unit (b) 32 thousands (c) 48 thousands (d) 56 thousands (e) The average cost per unit has no minimum

Explanation / Answer

C(q) = 0.9q2 +2q +810

Average cost = C(q) /q = (0.9q2 +2q +810)/q = 0.9q + 810/q + 2

minimum average cost ==> C '(q) = 0

==> 0.9 - 810/q2 = 0

==> q2 = 810/0.9 = 900

==> q = 30

Hence average cost is minimized at a level production equal to 30 thousands

2) minimum average cost = 0.9(30) + 810/30 + 2 = 27 + 27 +2 = 56 thousands

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