Which one of the following set of constraints depicted the shaded region given below?
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → \( 2x+y\le 50,\ x+2y\le 40,\ x\ge 0,\ y\ge 0 \)
The two boundary lines are read from their intercepts.
Line \( l_2 \) passes through (0, 50) and C(25, 0):
\( \displaystyle \frac{x}{25}+\frac{y}{50}=1\ \Rightarrow\ 2x+y=50 \)
Line \( l_1 \) passes through A(0, 20) and (40, 0):
\( \displaystyle \frac{x}{40}+\frac{y}{20}=1\ \Rightarrow\ x+2y=40 \)
The shaded region OABC lies in the first quadrant and contains the origin O(0, 0). Testing O in each line:
\( \displaystyle 2(0)+0=0\le 50\ \Rightarrow\ 2x+y\le 50 \)
\( \displaystyle 0+2(0)=0\le 40\ \Rightarrow\ x+2y\le 40 \)
Together with \( x\ge 0,\ y\ge 0 \), this is Option (3).