Question

The maximum value of the objective function $z = 2x + 3y$ subject to the constraints $x + y \le 5$, $2x + y \ge 4$ and $x \ge 0, y \ge 0$ is

• A. 15
• B. 10
• C. 20
• D. 25

Hint:

Look closely at the objective function $z = 2x + 3y$. The weighting coefficient for $y$ ($3$) is larger than for $x$ ($2$). To maximize $z$ quickly when the constraints allow, test the points that lie highest along the vertical $y$-axis first. Comparing $(0,5)$ and $(0,4)$ immediately yields the maximum value of $15$.

Answer 1

The Correct Option is A

Step 1: Understanding the Question:
The problem presents a linear programming problem (LPP). We need to find the maximum value of the objective function $z = 2x + 3y$ inside the feasible region bounded by the given linear inequalities.

Step 2: Key Formula or Approach:
According to the Corner Point Theorem, the optimal (maximum or minimum) value of an objective function in a linear programming problem always occurs at one of the corner vertices of the bounded feasible region. We must identify the corner points by plotting the lines and checking their intersections.

Step 3: Detailed Explanation:
1. Find the intercepts and boundary lines for the constraints: For line 1: $x + y = 5$. Intercepts are $(5,0)$ and $(0,5)$. The inequality is $x + y \le 5$, which covers the region towards the origin. For line 2: $2x + y = 4$. Intercepts are $(2,0)$ and $(0,4)$. The inequality is $2x + y \ge 4$, which covers the region away from the origin. Non-negativity constraints $x \ge 0, y \ge 0$ limit the region to the first quadrant. 2. Identify the corner vertices of the resulting shaded feasible region: $A(0, 4)$ (y-intercept of line 2) $B(2, 0)$ (x-intercept of line 2) $C(5, 0)$ (x-intercept of line 1) $D(0, 5)$ (y-intercept of line 1) 3. Compute the value of the objective function $z = 2x + 3y$ at each corner point: At $A(0, 4)$: $z = 2(0) + 3(4) = 12$ At $B(2, 0)$: $z = 2(2) + 3(0) = 4$ At $C(5, 0)$: $z = 2(5) + 3(0) = 10$ At $D(0, 5)$: $z = 2(0) + 3(5) = 15$ 4. Compare the values: the maximum value obtained is 15, which occurs at the vertex $D(0, 5)$.

Step 4: Final Answer:
The maximum value of the objective function is 15, matching option (A).