The cost of emissions at a form is given by C(q) = 6,000 + 100g^2 where q is the

Question

The cost of emissions at a form is given by C(q) = 6,000 + 100g^2 where q is the reduction in emissions (in pounds of pollutant per day) and C is the daily cost to the from (in dollars) of this reduction. Government clean-air subsidies amount to $500 per pound of polutant removed. How many pounds of pollutant should the firm remove each day in order to minimize net cost (cost minus subside)? ibs of pollutant Hercules Films is deciding on the price of the video releaseof its firm Bride of the San of Fransestem. Marketing estimates that at a price of p dollars, it can sell q = 300,000 - 10,000 p coples, but each copy costs $4 to make. What price will give the greastes profit? p = $ The demand equation for your company's virtual reality video headsets is p = 2,000/q^ 3 Where q is the total number of

Explanation / Answer

1 Net cost = 6000 + 100q2 - 500q

   To minimize it, we set the first derivative to zero.

C(q) = 200q - 500 = 0 => q = 2.5 pounds/day

2 Profit = Sales - Cost

         = Price * Quantity Sold - Cost/copy* qty sold

         = (300,000 - 10,000p)p - 4(300,000 - 10,000p) = 260,000p - 1,200,000 - 10,000p2

   Set the derivative to zero. That is, 260,000 - 20,000p = 0 which gives p = $13

3 a) Profit = Sales - Cost = pq - 120q

     = 2000q/ q.3 - 120q = 2000q.7 - 120q

    Taking 1st derivative, 1400/q.3 - 120 = 0, or, q.3 = 35/3

   Solving, q = 3601 (ans to part b)

   p = 2000/3601.03 = $171.44 (ans to part c)

Total profit = 3601 (171.44 -120) = $185,222

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