The number of feasible solutions of the linear programming problem given as Maximize $z=15x+30y$ subject to constraints : $3x+y\le12, x+2y\le10, x\ge0, y\ge0$ is
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Step-by-Step Solution
First, graph the inequalities:
$3x + y \le 12$
$x + 2y \le 10$
$x \ge 0$
$y \ge 0$
Find the intersection points of the lines:
$3x + y = 12$ and $x + 2y = 10$
Multiply the first equation by 2: $6x + 2y = 24$
Subtract the second equation: $5x = 14$, so $x = \frac{14}{5} = 2.8$
Substitute $x$ into the second equation: $2.8 + 2y = 10$, so $2y = 7.2$, and $y = 3.6$
Intersection point: $(2.8, 3.6)$
$3x + y = 12$ and $x = 0$: $(0, 12)$
$3x + y = 12$ and $y = 0$: $(4, 0)$
$x + 2y = 10$ and $x = 0$: $(0, 5)$
$x + 2y = 10$ and $y = 0$: $(10, 0)$
$x = 0$ and $y = 0$: $(0, 0)$
Identify the feasible region. The vertices of the feasible region are $(0, 0)$, $(4, 0)$, $(0, 5)$, and $(2.8, 3.6)$.
Since the feasible region is a polygon, and any point within the region is a feasible solution, there are infinitely many feasible solutions.
Correct Answer: infinite
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Pedagogical Audit
Bloom's Analysis:
This is an APPLY question because it requires students to apply their understanding of linear programming constraints and feasible regions to determine the number of feasible solutions. They need to use the given inequalities to define the feasible region and then count the solutions within that region.
Knowledge Dimension:PROCEDURAL
Justification:The question requires the student to execute a procedure, namely, graphing the inequalities, identifying the feasible region, and then determining the nature of solutions (finite or infinite).
Syllabus Audit:
In the context of CBSE Class 12, this is classified as KNOWLEDGE. The question directly assesses the student's understanding and application of linear programming concepts as taught in the textbook.