Adam Goodman determines that when q thousand units of his products are produced

Question

Adam Goodman determines that when q thousand units of his products are produced each month, they will all be sold at a price of p(q) = 22.2 - 1.2q dollars per unit. The total cost of producing the q units will be C(q) = 0.4q^2 + 3q + 40 thousand dollars. a) how many units should Adam produce to maximize profit? What is the maximum profit he can expect? b) how many units should he produce to minimize the average cost per unit production A(q) = C(q)/ q? What is the minimal cost?

Explanation / Answer

profit for q thousand units = selling price - cost price => p = (22.2 - 1.2q)(1000)(q) - (0.4q^2 + 3q + 40)(1000) => p = 22200q - 1200q^2 - 400q^2 - 3000q - 40000 => p = -1600q^2 + 19200q - 40000 maximaizing profit.. p' = -3200q + 19200 = 0 => q = 192/32 = 6 hence no. of units to be made for maximum profit in a month = 6000 => maximum profit = -1600*36 + 19200*6 - 40000 = 17600 = answer

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