Linear Programming Question 108
Question: Maximize $ Z=3x+5y, $ subject to $ x+4y\le 24, $
$ 3x+y\le 21, $
$ x+y,\le 9, $
$ x\ge 0,y\ge 0, $ is
Options:
A) $ 20at(1,0) $
B) $ 30at(0,6) $
C) $ 37at(4,5) $
D) $ 33at(6,3) $
Show Answer
Answer:
Correct Answer: C
Solution:
We have, maximize $ Z=3x+5y $ subject to constraints: $ x+4y\le 24,3x+y\le 21,x+y\le 9,x\ge 0,y\ge 0 $ Let $ {\ell_1}:x+4y=24 $
$ {\ell_2}:3x+y=21 $
$ {\ell_3}:x+y=9 $
$ {\ell_4}:x=0 $ and $ {\ell_5}:y=0 $ On solving these equations we will get points as O (0, 0), A (7, 0), B (6, 3), C (4, 5), D (0, 6) Now maximize $ Z=3x+5y $
$ ZatO(0,0)=3(0)+5(0)=0 $
$ ZatA(7,0)=3(7)+5(0)=21 $
$ ZatB(6,3)=3(6)+5(3)=33 $
$ ZatC(4,5)=3(4)+5(5)=37 $
$ ZatD(0,6)=3(0)+5(6)=30 $ Thus, Z is maximized at C (4, 5) and its maximum value is 37.
BETA
