The month of September is celebrated as the Rashtriya Poshan Maah across the country. Following a healthy and well-balanced diet is crucial in order to supply the body with the proper nutrients it needs. A balanced diet also keeps us mentally fit and promotes improved level of energy.
A dietician wishes to minimize the cost of a diet involving two types of foods, food X (x kg) and food Y (y kg) which are available at the rate of ₹ 16/kg and ₹ 20/kg respectively. The feasible region satisfying the constraints is shown in Figure-2.
On the basis of the above information, answer the following questions :
(i) Identify and write all the constraints which determine the given feasible region in Figure-2. (2)
(ii) If the objective is to minimize cost Z = 16x + 20y, find the values of x and y at which cost is minimum. Also, find minimum cost assuming that minimum cost is possible for the given unbounded region. (2)
Show answer & solution
- (i) The shaded region lies on the side away from the origin of the lines x + 2y = 10, x + y = 6 and 3x + y = 8, in the first quadrant.
- Constraints: , , , , .
- (ii) Corner points: D(0, 8): Z = 160; C(1, 5): Z = 16 + 100 = 116; B(2, 4): Z = 32 + 80 = 112; A(10, 0): Z = 160.
- Smallest value is 112 at B(2, 4). The half-plane 16x + 20y < 112, i.e. 4x + 5y < 28, has no point in common with the feasible region, so 112 is the minimum.
- Minimum cost ₹ 112 when x = 2 kg and y = 4 kg.