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Step 2 Formulation We are given that the costs are of two types fixed and variable The fixed costs are independent of the number of units produced e g rent and rates while the variable costs increase with the number of units produced e g material Initially we assume that the variable costs are directly proportional to the number of units produced this should simplify our model The company earn a certain amount of money by selling its products and wants to ensure that it is maximum For convenience we assume that all units produced are sold immediately The mathematical model Let x number of units produced and sold C total cost of production in rupees I income from sales in rupees P profit in rupees Our assumptions above state that C consists of two parts i fixed cost a in rupees ii variable cost b rupees unit produced Then C a bx Also income I depends on selling price s rupees uni Thus I Sx The profit P is then the difference between income and costs So s b x a independent dependent 3 We now have a mathematical model of the relationships 1 to 3 between the variables x C I P a b s These variables may be classified as Rationalised 2023 24 ot to be published X C I P parameters a b s The manufacturer knowing x a b s can determine P Step 3 From 3 we can observe that for the break even point i e make neither profit a nor loss he must have P 0 i e x units s b Steps 4 and 5 In view of the break even point one may conclude that if the company produces few units i e less than x units then the company will suffer loss s b a dP dx 1 s b MATHEMATICAL MODELLING a s b and if it produces large number of units i e much more than units then it can make huge profit Further if the break even point proves to be unrealistic then another model could be tried or the assumptions regarding cash flow may be modified Remark From 3 we also have 205 This means that rate of change of P with respect to x depends on the quantity h which is the difference of selling price and the variable cost of each product

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