Question

The minimum and maximum values of the objective function \(Z=3x+9y\), subject to \(x+3y\leq 60,x+y\geq 10,x\leq y,x\geq 0,y\geq 0\) are respectively...

• A. \(90\) and \(190\)
• B. \(60\) and \(180\)
• C. \(30\) and \(120\)
• D. \(40\) and \(220\)

Hint:

Find the corner points of the feasible region and evaluate Z at each.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept
The extreme values of Z occur at corner points of the feasible region bounded by the given lines.

Step 2: Corner points
\(x=y\) with \(x+y=10\): \((5,5)\).
\(x=y\) with \(x+3y=60\): \((15,15)\).
\(x=0\) with \(x+y=10\): \((0,10)\).
\(x=0\) with \(x+3y=60\): \((0,20)\).

Step 3: Evaluate Z
\(Z(5,5)=60\), \(Z(15,15)=180\), \(Z(0,10)=90\), \(Z(0,20)=180\).

Step 4: Compare
The minimum is 60 and the maximum is 180.

Final Answer:
The minimum is 60 and the maximum is 180. \[ \boxed{\text{(B) }60\text{ and }180} \]