Answer to Question #350139 in Calculus for Vickie

Question

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

Answer 1

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 = $\$3\,400\,000$