Question

Consider the following linear programming problem:

Maximize	$5X+6Y$
Subject to:	$4X+2Y\le 420$
	$1X+2Y\le 120$
	all variables $\ge 0$

Which of the following points $(X,Y)$ is in the feasible region?

• A. $(30,60)$
• B. $(105,0)$
• C. $(0,210)$
• D. $(100,10)$
• E. None of the above

Answer 1

Answer: (B), (D)

The correct options are
B $(105,0)$
D $(100,10)$

Feasible points are the points that satisfy the constraints.

Therefore, substitute the options in the constraint equations and verify.

$A.$ substituting $(30,60)$ in $4x+2y\le 420$

we get $4\times 30+2\times 60\le 420$

$⟹120+120\le 420⟹240\le 420$ True

substituting $(30,60)$ in $1x+2y\le 120$

we get $1\times 30+2\times 60\le 120$

$⟹30+120\le 120⟹150\le 120$ False

$B.$ substituting $(105,0)$ in $4x+2y\le 420$

we get $4\times 105+2\times 0\le 420$

$⟹420+0\le 420⟹420\le 420$ True

substituting $(105,0)$ in $1x+2y\le 120$

we get $1\times 105+2\times 0\le 120$

$⟹105+0\le 120⟹105\le 120$ True

$C.$ substituting $(0,210)$ in $4x+2y\le 420$

we get $4\times 0+2\times 210\le 420$

$⟹0+420\le 420⟹420\le 420$ True

substituting $(0,210)$ in $1x+2y\le 120$

we get $1\times 0+2\times 210\le 120$

$⟹0+240\le 120⟹240\le 120$ False

$D.$ substituting $(100,10)$ in $4x+2y\le 420$

we get $4\times 100+2\times 10\le 420$

$⟹400+20\le 420⟹420\le 420$ True

substituting $(100,10)$ in $1x+2y\le 120$

we get $1\times 100+2\times 10\le 120$

$⟹100+20\le 120⟹120\le 120$ True

Therefore option B and D are the points in the feasible region.