Question

The minimum value of \(z = 3x+5y\), subject to constraints \(x\leq 80\), \(y\geq 60\), \(x+y\leq 200\) & \(x,y\geq 0\) occurs at the point...

• A. \((0,200)\)
• B. \((60,0)\)
• C. \((0,60)\)
• D. \((80,60)\)

Hint:

Find the corner points of the feasible region and compare z at each.

Answer 1

The Correct Option is C

Step 1: Feasible region
The constraints are \(x\le80\), \(y\ge60\), \(x+y\le200\), \(x,y\ge0\). The corner points are \((0,60)\), \((80,60)\), \((80,120)\) and \((0,200)\).

Step 2: Evaluate z
\((0,60): 300\), \((80,60): 240+300 = 540\), \((80,120): 240+600 = 840\), \((0,200): 1000\).

Step 3: Minimum
The smallest value is \(300\) at \((0,60)\). Option (C). The point \((60,0)\) is not feasible because \(y\ge60\) fails.

Final Answer:
The minimum occurs at (0, 60). \[ \boxed{\text{(C)}\ (0,60)} \]