Question

The minimum value of $Z=4x+3y$ subjected to the constraints $3x+2y\ge 160,5x+2y\ge 200,x+2y\ge 80;x,y\ge 0$ is

• A. $220$
• B. $300$
• C. $230$
• D. none of these

Answer 1

Answer: (A)

Let $l_1:3x+2y=160,l_2:5x+2y=200$
$l_3:x+2y=80,l_4:x=0,l_5:y=0$

For B : Solving $l_1$ and $l_3$, we get $B(40,20)$
For C : Solving $l_1$ and $l_2$, we get $C(20,50)$
Shaded region is the feasible region,
where $A(80,0),B(40,20),C(20,50),D(0,100)$
Now minimize $Z=4x+3y$
$Z$ at $A(80,0)=320$
$Z$ at $B(40,20)=40\times 4+20\times 3=220$
$Z$ at $C(20,50)=20\times 4+50\times 3=230$
$Z$ at $D(0,100)=300$
Minimum value of $Z$ is $220$.