A manufacturer of machinery parts determines that q units of a particular piece
ID: 2845572 • Letter: A
Question
A manufacturer of machinery parts determines that q units of a particular piece will be sold when the price is p=190?q dollars per unit. The total cost of producing those q units is C(q) dollars, where C(q)=q^3-15q^2+6q+4000
a)how much profit is derived from the sale of the q units at p dollar per unit?
P(q)= dollars
(b) For what value of q is profit maximized? Round to the nearest whole number.
q= units
(c)Find the consumers' surplus for the level of production q0 that corresponds to maximum profit.
CS= dollarsC(q)=q3
Explanation / Answer
a)
p = 190 - q
Revenue R = price*q = (190-q)*q = 190*q - q^2
Profit = Revenue - cost = (190q - q^2) - (q^3-15q^2+6q+4000) = -q^3 + 14q^2 + 184q - 4000
b)
dC / dq = 3q^2 - 30q + 6
dR / dq = 190 - 2q
For max. profit, we equate both the expressions.
3q^2 - 30q + 6 = 190 - 2q
3q^2 - 28q - 184 = 0
q = 13.78 (almost 14)
c)
Consumer surplus = Integral (q = 0 to q = 14) (190 - q) dq
= (q = 0 to q = 14) [190q - q^2 /2]
= 190*14 - 14^2 /2
= 2562
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