1. Find the formulae of the linear functions such that:

2. The total cost, £C, of manufacturing a batch of articles is suspected to be related to the number of articles in the batch, x, by a linear model of the form C = ax + b where a and b are constants. The following returns were noted:

(a) By plotting the graph of C against x show that these figures fit the above model and find the values of a and b. What is the significance of a and b in this problem?

(b) The selling price is fixed at £5.30 per article. Using the same axes and scales, plot another graph relating the total selling price, £S, to the total number of articles sold.
From your graphs determine

3. If the retail price of a particular item produced by a firm is £P, then the number of items the firm is prepared to supply at this price is given by the equation

whilst the number D of items that customers will buy at this price is given by the equation

(a) Using the same axes draw graphs of D and S against P. Indicate the point corresponding to the situation when the supply S equals the demand D (that is, the market is in equilibrium) and determine the number of articles produced in this case. Find also the selling price corresponding to the equilibrium state of the market.

(b) When the government levies a tax of £10 per item sold, the customers are concerned only with the new total retail price £P, whereas the suppliers react as though the price was £(P − 10).

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(a) y = 0 where x = 32, y = 100 where x = 220 (b) y = 8 where x = 10, y = 12 where x = 25