Q1478: In the Linear Programming Problem for objective function $Z=...

Class CBSE Class 12 Mathematics Linear Programming Q #1478

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3 Marks 2025 AISSCE(Board Exam) SA

In the Linear Programming Problem for objective function $Z=18x+10y$ subject to constraints $4x+y\ge20$, $2x+3y\ge30$, $x,y\ge0$ find the minimum value of Z.

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Detailed Solution

Step 1: Graph the Constraints

First, we need to graph the inequalities to find the feasible region. The constraints are: $4x + y \ge 20$ $2x + 3y \ge 30$ $x \ge 0$ $y \ge 0$

Step 2: Find the Intersection Points

Find the intersection points of the lines: $4x + y = 20$ and $2x + 3y = 30$ Multiply the first equation by 3: $12x + 3y = 60$ Subtract the second equation from the modified first equation: $(12x + 3y) - (2x + 3y) = 60 - 30$ $10x = 30$ $x = 3$ Substitute $x = 3$ into the first equation: $4(3) + y = 20$ $12 + y = 20$ $y = 8$ So, the intersection point is $(3, 8)$. The intersection of $4x+y=20$ with $x=0$ is $(0,20)$. The intersection of $4x+y=20$ with $y=0$ is $(5,0)$. The intersection of $2x+3y=30$ with $x=0$ is $(0,10)$. The intersection of $2x+3y=30$ with $y=0$ is $(15,0)$.

Step 3: Identify Corner Points of the Feasible Region

The corner points of the feasible region are $(0, 20)$, $(3, 8)$, and $(15, 0)$.

Step 4: Evaluate the Objective Function at the Corner Points

Evaluate $Z = 18x + 10y$ at each corner point: At $(0, 20)$: $Z = 18(0) + 10(20) = 0 + 200 = 200$ At $(3, 8)$: $Z = 18(3) + 10(8) = 54 + 80 = 134$ At $(15, 0)$: $Z = 18(15) + 10(0) = 270 + 0 = 270$

Step 5: Determine the Minimum Value

The minimum value of $Z$ is $134$ at the point $(3, 8)$.

Final Answer: 134

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Pedagogical Audit

Bloom's Analysis: This is an REMEMBER question because it requires recalling the steps to solve a linear programming problem, including graphing constraints, finding intersection points, and evaluating the objective function at corner points.

Knowledge Dimension: PROCEDURAL

Justification: The question requires the student to apply a specific algorithm or method (linear programming) to find the minimum value of the objective function. This involves a series of steps and techniques.

Syllabus Audit: In the context of CBSE Class 12, this is classified as KNOWLEDGE. It is a standard problem-solving exercise directly related to the textbook content on linear programming.