Question

The feasible region of an LPP is shown in the figure below.
If \( z=3x+9y\), then the minimum value of \(z\) occurs at :

• A. \((0,10)\)
• B. \((0,20)\)
• C. \((5,5)\)
• D. \((15,15)\)

Answer 1

The Correct Option is C

The given problem involves finding the minimum value of the function \( z = 3x + 9y \) over a feasible region defined by an LPP (Linear Programming Problem). The coordinates provided represent the vertices of this feasible region. We need to calculate the value of \( z \) at each vertex to identify where the minimum occurs.

1. Calculate \( z \) at (0,10):
   \( z = 3(0) + 9(10) = 90 \)
2. Calculate \( z \) at (0,20):
   \( z = 3(0) + 9(20) = 180 \)
3. Calculate \( z \) at (5,5):
   \( z = 3(5) + 9(5) = 60 \)
4. Calculate \( z \) at (15,15):
   \( z = 3(15) + 9(15) = 180 \)

From these calculations, the minimum value of \( z \) is 60, which occurs at the point (5,5). Therefore, (5,5) is the correct answer.