Question

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.
\includegraphics[width=\linewidth]{latex1.png} A dietician wishes to minimize the cost of a diet involving two types of foods, food \( X \) (in kg) and food \( Y \) (in kg), which are available at the rate of \( \text{₹} 16/\text{kg} \) and \( \text{₹} 20/\text{kg} \), 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. [(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 the minimum cost assuming that minimum cost is possible for the given unbounded region.

Hint:

For linear programming problems, always identify the constraints based on the inequalities that bound the feasible region in the graph. These constraints are essential for solving optimization problems.

Answer 1

(i) Constraints determining the feasible region:
From the given Figure-2, the feasible region is bounded by the following constraints: \[ 3x + y \leq 8, \quad x + y \geq 4, \quad 4x + 5y = 28, \quad 2x + y \geq 10, \quad x \geq 0, \quad y \geq 0. \] Solution:
The constraints are: \[ \boxed{3x + y \leq 8, \, x + y \geq 4, \, 4x + 5y = 28, \, 2x + y \geq 10, \, x \geq 0, \, y \geq 0.} \]