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22 December, 04:44

A firm is planning to manufacture a new product. As the selling price is increased, the quantity that can be sold decreases. Numerically they estimate

P = $35.00 - 0.02Q

(P = selling price per unit, Q = quantity sold per year)

On the other hand, management estimates that the average cost of manufacturing and selling the product will decrease as the quantity sold increases

C = $4.00Q + $8000

where C = cost to produce and sell Q per year

The want to maximize profit. What quantity should the decision makers plan to produce and sell each year?

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Answers (1)
  1. 22 December, 06:51
    0
    Profit Maximising Quantity = 775

    Explanation:

    Price P = 35 - 0.02Q

    Total Revenue TR = Price x Quantity = P X Q

    = (35 - 0.02Q) (Q) = 35Q - 0.02Q^2

    Total Cost TC = 8000 + 4Q

    Profit = TR - TC

    [35Q - 0.02Q^2] - [8000+4Q] = 35Q - 0.02Q^2 - 8000 - 4Q

    Profit Function = - 0.02Q^2 + 31Q - 8000

    To find out profit maximising Quantity, we will differentiate Profit Function with respect to Q & equate it to 0.

    dTR / dQ = - 0.04Q + 31 = 0

    Q = 31/0.04 = 775

    To verify whether 775 is profit maximising Q, we will do second derivative & check that it is negative.

    d^2TR / dQ^2 = - 0.04 i. e < 0 (negative)

    So 775 is profit maximising quantity
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