For a L.P.P , Maximize \( z=3x+5y \) subject to the constraints \( x+2y\le 20 \), \( x+y\le 15 \), \( y\le 5 \), \( x,y\ge 0 \).
Which of the following statements are correct ?
(A). The feasible region is bounded.
(B). The feasible region is unbounded.
(C). The maximum of z occurs at (10,5).
(D). The maximum of z occurs at (20,0).
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → (A) and (C) only
The lines \( x+y=15 \) and \( y=5 \) meet at (10, 5). This point also satisfies \( x+2y=20 \), so that line only touches the region there.
The other corners are (0, 0), (15, 0) and (0, 5), so the feasible region is a closed polygon and is bounded. At the corners,
\( z(0,0)=0 \)
\( z(15,0)=45 \)
\( z(10,5)=55 \)
\( z(0,5)=25 \)
So the feasible region is bounded, and the maximum of z is 55, at (10, 5).
Checking the statements
(A) The feasible region is bounded. — Correct. It is the polygon with corners (0, 0), (15, 0), (10, 5) and (0, 5).
(B) The feasible region is unbounded. — Incorrect. The region is that closed polygon.
(C) The maximum of z occurs at (10,5). — Correct. \( z(10,5)=55 \), the largest corner value.
(D) The maximum of z occurs at (20,0). — Incorrect. (20, 0) is not feasible, since \( 20+0\gt 15 \).