Answer to Question #350139 in Calculus for Vickie

Question #350139

The demand function is Q=280000-400p

Where Q equals the number of units demanded and p equals the price in dollars of the total cost of producing Q units of the product is estimated by the function

C=350000+300q+0.0015q2

I) how many units should Q produce in order to maximize its annual profit

Ii) what price should be charged

III) determine the annual profit expected


1
Expert's answer
2022-06-15T05:53:19-0400

Demand Function, "Q = 280\\,000-400p"

Inverse Demand Function, "P = \\frac{280,000-Q}{400} = 700 - \\frac{Q}{400}"


"\\text{Total Revenue} = \\text{Price} \\cdot \\text{Quantity} = 700-\\frac{Q}{400}\\cdot Q = 700Q-\\frac{Q^2}{400}"


"\\text{Marginal revenue function} = \\frac{d}{dQ}(700Q-\\frac{Q^2}{400})=700-0.005Q"


"\\text{Total Cost function} = 350\\,000+300Q+0.0015Q^2"


"\\text{Marginal Cost} = \\frac{d}{dQ}(350\\,000+300Q+0.0015Q^2)=300+0.003Q"


Profit is maximized when Marginal Cost = Marginal Revenue (Price):

"300+0.003Q = 700-0.005Q"

"0.003Q+0.005Q = 700-300"

"0.008Q=400"

"Q=\\frac{400}{0.008} = 50\\,000"


i) The firm should produce "50\\,000" units to maximize its profit


ii) Price that should be charged = Marginal Revenue at "50\\,000" units of output = "\\$(700 -0.005(50\\,000)) = \\$(700 - 250) = \\$ 450"


iii)At maximum profit condition, "Q = 50\\,000"

"\\text{Total Revenue} = 700Q-0.005Q^2 = 700 x 50\\,000 - 0.005 \\cdot 50\\,000^2 = \\$22\\,500\\,000"

"\\text{Total Cost} = 350\\,000+300Q+0.0015Q^2 =" "= 350\\,000+300 \\cdot 50\\,000 + 0.0015 \\cdot 50\\,000^2 =\\$19\\,100\\,000"

"\\text{Annual Profit} = \\text{Total Revenue} - \\text{Total Cost} = \\$22\\,500\\,000-\\$19\\,100\\,000 = \\$3\\,400\\,000"

Therefore, expected annual profit = "\\$3\\,400\\,000"



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