Q15 of 17 Page 12

There are two types of fertilizers ‘A’ and ‘B’. ‘A’ consists of 12% nitrogen and 5% phosphoric acid whereas ‘B’ consists of 4% nitrogen and 5% phosphoric acid. After testing the soil conditions, the farmer finds that he needs at least 12 kg of nitrogen and 12 kg of phosphoric acid for his crops. If ‘A’ costs ₹ 10 per kg and ‘B’ cost ₹ 8 per kg, then graphically determine how much of each type of fertilizer should be used, so that nutrient requirements are met at a minimum cost. [CBSE 2016]

Given: Fertilizer ‘A’ consists of 12% nitrogen and 5% phosphoric acid whereas Fertilizer ‘B’ consists of 4% nitrogen and 5% phosphoric acid. ‘A’ costs Rs. 10 per kg and ‘B’ cost Rs. 8 per kg


To find: quantity of fertilizer should be used, so that nutrient requirements are met at a minimum cost


Let the quantity of fertilizer A and B be used as x and y respectively, and the total cost be z


⇒ z = 10x + 8y


We need to minimize the cost


Hence, the mathematical formulation of LPP is


Minimize z = 10x + 8y


subject to the constraints,











The feasible region determined by the system of constraints is as follows:



The corner points of the enclosed region are A (0, 300), B(30, 210), C(240, 0)


The value of z at these corners points is as follows:


Case 1: A(0, 300)


z = 10x + 8y


⇒ z = 10(0) + 8(300)


⇒ z = 0 + 2400


⇒ z = 2400


Case 2: B(30, 210)


z = 10x + 8y


⇒ z = 10(30) + 8(210)


⇒ z = 300 + 1680


⇒ z = 1980


Case 3: C(240, 0)


z = 10x + 8y + 370


⇒ z = 10(240) + 8(0)


⇒ z = 2400 + 0


⇒ z = 2400


The value of z is minimum in the second case at point B(30, 210)


Hence, the quantity of fertilizer A and B be used as 30 Kg and 210 Kg respectively, and the total cost is Rs. 1980


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A retired person wants to invest an amount of ₹ 50,000. His broker recommends investing in two types of bonds ‘A’ and ‘B’ yielding 10% and 9% return respectively on the invested amount. He decides to invest at least ₹ 20,000 in bond ‘A’ and at least ₹ 10,000 in bond ‘B’. He also wants to invest at least as much in bond ‘A’ as in bond ‘B’. Solve this linear programming problem graphically to maximize his returns. [CBSE 2016]

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A factory manufactures two types of screws A and B, each type requiring the use of two machines, an automatic and a hand – operated. It takes 4 minutes on the automatic and 6 minutes on the hand operated machines to manufacture a packet of screws ‘B’. Each machine is availble for at most 4 hours on any day. The manufacturer can sell a packet of screws ‘A’ at a profit of 70 paise and screws ‘B’ at a profit of Rs. 1. Assuming that he can sell all the screws he manufactrures, how many packets of each type should the factory owner produce in a day in order to maximize his profit ? Formulate the above LPP and solve it graphically and find the maximum profit.[CBSE 2018]

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A company produces two types of goods, A and B, that require gold and silver. Each unit of type A requires 3 g of silver and 1 g of gold while that of B requires 1 g of silver and 2 g of gold. The company can use at most 9 g of silver and 8 g of gold. If each unit of type A brings a profit of Rs40 and that of type B Rs50, find the number of units of each type that the company should produce to maximize the profit. Formulate and solve graphically the LPP and find the maximum profit.[CBSE 2018]