The costs of a firm which makes climbing boots are of two kind:
-Fixed cost (plant rates, office expenses): $2000 per week;
-Production costs (material, labour): $20 for each pair of boots made.
Market research suggest that if they price the boots at $30 a pair they will sell 500 pairs a week, but that at $55 a apair they will sell none at all; and between these values the graph of sales against profit is a straight line.
If they price boots at $x pair (30<x<55) (inclusive) find expressions for
a) weekly sales,
b) the weekly reciepts
c) the weekly costs
assuming that just enought boots are made
hence showing that the weeklly profit, $P, is given by
P=-20x2+ 1500x-2400.
Find the price at which the boots should be sold to maximise the profit.