The feasible region of an LPP is a bounded region by vertices (4, 18), (6, 12), (40, 0) and (15, 0). Then the point (15, 15) lies in the
A. open half plane of \( x+2y\lt40 \).
B. closed half plane of \( 3x+y\ge30 \).
C. closed half plane of \( 4x+3y\ge60 \).
D. feasible region of the given LPP.
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → B and C only
Each side of the region joins two consecutive vertices, so the sides lie on the lines
\( \displaystyle (4,18),(6,12):\ 3x+y=30 \)
\( \displaystyle (6,12),(15,0):\ 4x+3y=60 \)
\( \displaystyle (4,18),(40,0):\ x+2y=40 \)
The region is therefore \( x+2y\le40,\ 3x+y\ge30,\ 4x+3y\ge60,\ y\ge0 \).
So at the point (15, 15): \( x+2y=45 \), \( 3x+y=60 \) and \( 4x+3y=105 \).
Checking the statements
(A) open half plane of \( x+2y\lt40 \) — Incorrect. \( x+2y=45 \), which is not less than 40.
(B) closed half plane of \( 3x+y\ge30 \) — Correct. \( 3x+y=60\ge30 \).
(C) closed half plane of \( 4x+3y\ge60 \) — Correct. \( 4x+3y=105\ge60 \).
(D) feasible region of the given LPP — Incorrect. \( x+2y=45\gt40 \), so the point violates \( x+2y\le40 \) and lies outside the region.