The costs of a firm which makes climbing boots are of two kinds: Fixed costs (pl

Question

The costs of a firm which makes climbing boots are of two kinds: Fixed costs (plant, rates, office expenses): 2000 per week; Production costs (materials, labor): 20 for each pair of boots made. Market research suggests that, if they price the boots at 30 a pair they will sell 500 pairs a week, but that at 55 a pair they will sell none at all: and between these values the graph of Hales against price is a straight line. If they price boots at x a pair (30 le x le 55) find expressions for the weekly sales, the weekly receipts, the weekly costs (assuming that just enough boots are made). Hence show that the weekly profit, p, is given by P = -20x2 + 1500x - 24 000. Find the price at which the boots should be sold to maximize the profit. Answers 1100 - 20x x(1100 - 20x) (24 000 - 400x); 37.50

Explanation / Answer

Suppose price as x and sales as y

a) the two points (x,y) given by (30,500) and (55,0)

we find the equation of the line for weekly sales i.e. y-0 + (500/25)(x-55) = 0 => y = 1100 - 20x (answer)

b) weekly reciept = x(1100 -20x) i.e. price * units sold

c) assume x as price and y as cost i.e (30,12000) and (55,2000)

we find the equation of the line for weekly costs i.e. y-55 + (10000/25)(x-55) = 0 => y = 24000-400x (answer)

profit(P) = sales - cost = x(1100 -20x) - (24000-24x) = -20x2 + 1500x - 24000 (answer)

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