Question

For the linear programming problem, \(x+2y\leq 10, 3x+y\leq 12, x,y\geq 0\), the maximum value of \(z = 5x+10y\) occurs at every point on the line segment joining the points..

• A. \((0,0)\) and \((4,0)\)
• B. \((0,0)\) and \((0,5)\)
• C. \((4,0)\) and \((\frac{14}{5},\frac{18}{5})\)
• D. \((0,5)\) and \((\frac{14}{5},\frac{18}{5})\)

Hint:

The objective is parallel to the constraint \(x+2y\le10\), so the maximum occurs along an edge.

Answer 1

The Correct Option is D

Step 1: Compare slopes
The objective line $5x+10y=c$ has the same slope as $x+2y=10$.

Step 2: Conclude
So the objective line coincides with that edge when $z=50$, and every point on that edge is optimal. The edge runs between $(0,5)$ and $\left(\tfrac{14}{5},\tfrac{18}{5}\right)$, option (D).

Final Answer:
Option (D) names the optimal segment. \[ \boxed{(0,5)\text{ to }\left(\dfrac{14}{5},\dfrac{18}{5}\right)} \]