Z at Q 20000 6000 8 20000 7 6000 202000 Z at R 10500 34500 8 x 10500 7 34500 325500 Z at S 5000 40000 8 5000 7 40000 320000 Z at T 0 40000 7 40000 280000 Now observe that the profit is maximum at x 10500 and y 34500 and the maximum profit is 325500 Hence the manufacturer should produce 10500 bottles of M medicine and 34500 bottles of M medicine in order to get maximum profit of 325500 Example 6 Suppose a company plans to produce a new product that incur some costs fixed and variable and let the company plans to sell the product at a fixed price Prepare a mathematical model to examine the profitability Solution Step 1 Situation is clearly identifiable 204 MATHEMATICS Rationalised 2023 24 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 unit Thus I Sx The profit P is then the difference between income and costs So s b x a ott be published independent dependent 1 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 2 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 units a nor loss he must have P 0 i e x s b Stens 4 and 5 In view of the break even point one may conclude that if the company