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=280000400pQ = 280\,000-400p

Inverse Demand Function, P=280,000Q400=700Q400P = \frac{280,000-Q}{400} = 700 - \frac{Q}{400}


Total Revenue=PriceQuantity=700Q400Q=700QQ2400\text{Total Revenue} = \text{Price} \cdot \text{Quantity} = 700-\frac{Q}{400}\cdot Q = 700Q-\frac{Q^2}{400}


Marginal revenue function=ddQ(700QQ2400)=7000.005Q\text{Marginal revenue function} = \frac{d}{dQ}(700Q-\frac{Q^2}{400})=700-0.005Q


Total Cost function=350000+300Q+0.0015Q2\text{Total Cost function} = 350\,000+300Q+0.0015Q^2


Marginal Cost=ddQ(350000+300Q+0.0015Q2)=300+0.003Q\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=7000.005Q300+0.003Q = 700-0.005Q

0.003Q+0.005Q=7003000.003Q+0.005Q = 700-300

0.008Q=4000.008Q=400

Q=4000.008=50000Q=\frac{400}{0.008} = 50\,000


i) The firm should produce 5000050\,000 units to maximize its profit


ii) Price that should be charged = Marginal Revenue at 5000050\,000 units of output = $(7000.005(50000))=$(700250)=$450\$(700 -0.005(50\,000)) = \$(700 - 250) = \$ 450


iii)At maximum profit condition, Q=50000Q = 50\,000

Total Revenue=700Q0.005Q2=700x500000.005500002=$22500000\text{Total Revenue} = 700Q-0.005Q^2 = 700 x 50\,000 - 0.005 \cdot 50\,000^2 = \$22\,500\,000

Total Cost=350000+300Q+0.0015Q2=\text{Total Cost} = 350\,000+300Q+0.0015Q^2 = =350000+30050000+0.0015500002=$19100000= 350\,000+300 \cdot 50\,000 + 0.0015 \cdot 50\,000^2 =\$19\,100\,000

Annual Profit=Total RevenueTotal Cost=$22500000$19100000=$3400000\text{Annual Profit} = \text{Total Revenue} - \text{Total Cost} = \$22\,500\,000-\$19\,100\,000 = \$3\,400\,000

Therefore, expected annual profit = $3400000\$3\,400\,000



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