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This means that rate of change of P with respect to x depends on the quantity s b which is the difference of selling price and the variable cost of each product Thus in order to gain profit this should be positive and to get large gains we need to produce large quantity of the product and at the same time try to reduce the variable cost Example 7 Let a tank contains 1000 litres of brine which contains 250 g of salt per litre Brine containing 200 g of salt per litre flows into the tank at the rate of 25 litres per minute and the mixture flows out at the same rate Assume that the mixture is kept uniform all the time by stirring What would be the amount of salt in the tank at any time t Solution Step 1 The situation is easily identifiable Step 2 Let y y t denote the amount of salt in kg in the tank at time t in minutes after the inflow outflow starts Further assume that y is a differentiable function When t 0 i e before the inflow outflow of the brine starts y 250 g x 1000 250 kg Note that the change in y occurs due to the inflow outflow of the mixture Now the inflow of brine brings salt into the tank at the rate of 5 kg per minute as 25 x 200 g 5 kg and the outflow of brine takes salt out of the tank at the rate of 25 or 206 00 4 1000 Thus the rate of change of salt with respect to t is given by dy Why dt or or 40 kg per minute as at time t the salt in the tank is MATHEMATICS Therefore dt 40 ax 1 1 40 e This gives a mathematical model for the given problem Step 3 Equation 1 is a linear equation and can be easily solved The solution of 1 is given by ye 200 e where c is the constant of integration Note that when t 0 y 250 Therefore 250 200 C or C 50 Then 2 reduces to y 200 50 1 40 t y 40 y 5 Rationalised 2023 24 y 200 50 e 40 200 Ce C or y t 50 1 40 e 40 1000 ERT kg 40 1 published 4

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