\(\text{The feasible region represented by the constraints } 4x + y \geq 80, \; x + 5y \geq 115, \; 3x + 2y \leq 150, \; x, y \geq 0 \; \text{of an LPP is:}\)
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When solving linear programming problems, plotting the constraints is a crucial step. Each constraint will represent a line, and the feasible region is the intersection of all regions that satisfy the inequalities. Remember, the feasible region must satisfy all constraints, and regions outside of it do not meet the required conditions. Always check the boundaries and verify that all constraints are satisfied to determine the correct feasible region.
- Region A
- Region B
- Region C
- Region D
The Correct Option is C
Approach Solution - 1
To determine the feasible region represented by the constraints, we follow these steps:
Plotting the Constraints:
The constraints are:
\[4x + y \geq 80\]
\[x + 5y \geq 115\]
\[3x + 2y \leq 150\]
\[x, y \geq 0 \text{ (indicating the feasible region is in the first quadrant)}\]
Each constraint represents a line in the \( xy \)-plane.
Identifying the Feasible Region:
The region satisfying all the constraints is the shaded region bounded by the intersection of the lines.
Based on the plot provided, Region C is enclosed by these lines and represents the feasible solution set for the linear programming problem (LPP).
Verification:
Region C satisfies all the constraints, including the inequality \( 3x + 2y \leq 150 \), which bounds it from above.
Other regions do not satisfy all the constraints simultaneously.
Thus, the feasible region for the given LPP is represented by Region C.
Approach Solution -2
To determine the feasible region represented by the constraints, we follow these steps:
Plotting the Constraints:
The constraints are:
\[ 4x + y \geq 80 \]
\[ x + 5y \geq 115 \]
\[ 3x + 2y \leq 150 \]
\[ x, y \geq 0 \text{ (indicating the feasible region is in the first quadrant)} \]
Each constraint represents a line in the \( xy \)-plane.
Identifying the Feasible Region:
The region satisfying all the constraints is the shaded region bounded by the intersection of the lines. Based on the plot provided, Region C is enclosed by these lines and represents the feasible solution set for the linear programming problem (LPP).
Verification:
Region C satisfies all the constraints, including the inequality \( 3x + 2y \leq 150 \), which bounds it from above. Other regions do not satisfy all the constraints simultaneously.
Thus, the feasible region for the given LPP is represented by Region C.
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