A firm is planning to manufacture a new product. As the selling price is increas

Question

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?

A detailed explanation would be great - will rate!

Explanation / Answer

These are the basic steps in the question: 1/Work out what the income is annually 2/Work out what the cost is annually 3/Work out what the profit is annually 4/Find the maximum profit (by differentiation) The question asks for the profit, which is the income minus the cost. The income is given by the number sold (Q) times the price (P), so: Income = PQ We know that: P = 35 - 0.02 Q Therefore: Income=PQ = 35Q - 0.02Q^2 The cost is given annually as: C=4Q+8000 So Cost = C Then we have: Income - Cost = Profit 35Q - 0.02Q^2 -4Q-8000 = Profit Lets call the profit A: -0.02Q^2 +31Q - 8000 = A We want to maximise the profit, A. A is at a maximum/minimum when A differentiated with respect to Q is equal to zero: dQ/dA= -0.04Q +31 = 0 Q=-31/-0.04=775 I'm sure he then knows how to go on to prove that this is a maximum rather than a minimum, by differentiating again.

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