Solve the linear programming problem graphically: Maximise Z = 4x + y subject to the constraints: x + y \(\le\) 50 3x + y \(\le\) 90 x \(\ge\) 0, y \(\ge\) 0
To Maximize Z = 4x + y ......(i) subject to the constraints: x + y \(\le\) 50 .....(ii) 3x + y \(\le\) 90 ......(iii) x \(\ge\) 0, y \(\ge\) 0 .....(iv) The shaded region in a figure is the feasible region determined by the system of constraints (ii) to (iv). We observe that the feasible region OABC is bounded. So, we now use Corner Point Method to determine the maximum value of Z. The coordinates of the corner points O, A, B and C are (0, 0), (30, 0), (20, 30) and (0, 50) respectively. Now we evaluate Z at each corner point.
Corresponding value of Z
Hence, maximum value of Z is 120 at the point (30, 0).
A dietician wishes to mix together two kinds of food X and Y in such a way that the mixture contains at least 10 units of vitamin A, 12 units of vitamin B and 8 units of vitamin C. The vitamin contents of one kg food is given below:
One kg of food X costs Rs 16 and one kg of food Y costs Rs 20. Find the least cost of the mixture which will produce the required diet?
Least cost of the mixture is Rs 122 (2 kg of Food X and 4 kg of food Y).
Least cost of the mixture is Rs 132 (3 kg of Food X and 4 kg of food Y).
Least cost of the mixture is Rs 112 (2 kg of Food X and 4 kg of food Y).
Least cost of the mixture is Rs 142 (2 kg of Food X and 5 kg of food Y).
A fruit grower can use two types of fertilizer in his garden, brand P and Q. The amounts (in kg) of nitrogen, phosphoric acid, potash, and chlorine in a bag of each brad are given in the table. Testes indicate that the garden needs at least 240 kg of phosphoric acid, at least 270 kg of potash and at most 310 kg of chlorine. If the grower wants to minimize the amount of nitrogen added to the garden, how many bags of each brand should be used? What is the minimum amount of nitrogen added in the garden?
A dietician has to develop a special diet using two foods P and Q. Each packet (containing 30 g) of food P contains 12 units of calcium, 4 units of iron, 6 units of cholesterol and 6 units of vitamin A, while each packet of the same quantity of food Q contains 3 units of calcium, 20 units of iron, 4 units of cholesterol and 3 units of vitamin A. The diet requires at least 240 units of calcium, atleast 460 units of iron and atmost 300 units of cholesterol. How many packets of each food should be used to maximize the amount of vitamin A in the diet? What is the maximum amount of vitamin A in the diet?