A manufacturer of machinery parts determines that q units of a particular piece

Question

A manufacturer of machinery parts determines that q units of a particular piece will be sold when the price is P = 150 - q dollars per unit. The total cost of producing those q units is C(q) dollars, where C(q) = q3 - 20q2 + 4q+ 10000 How much profit is derived from the sale of the q units at P dollar per unit? P(q) = dollars For what value of q is profit maximized? Round to the nearest whole number. q = units Find the consumers' surplus for the level of production q0 that corresponds to maximum profit. CS = dollars

Explanation / Answer

Revenue is pq= 150q- q^2

P(q)= 150q-q^2 -(q^3 -20q^2 +4q +1000)
P(q)= -q^3 +19q^2 +146q -1000

Profit is maxed where P'(q)=0 (derivative)
P'(q)= -3q^2 +38q +146
3q^2 -38q -146=0
q= (38± 38^2 +4*3*146)/6

q=( 38 +56.53)/6 16 (rounded to nearest whole #)

p is 134 at that quantity.

Consumer surplus is the area below the demand curve at (16,134) which is (150-134)*16 *1/2= 128

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