Question

If $x\ge 0$

$3y-2x\ge -12$
$2x+5y\le 20$
The area of the triangle formed in the xy plane by the system of inequalities above is:

• A. $60$
• B. $30$
• C. $40$
• D. $50$

Answer 1

The correct option is B $30$

given, $x\ge 0$

$3y-2x\ge -12$ and $2x+5y\le 20$

first, draw the graph for equations $x=0$

$3y-2x=-12$ and $2x+5y=20$

$x=0$ is Y-axis. Hence $x\ge 0$ includes the above region of the line.

for $3y-2x=-12$

substitute y=0 we get, $-2x=-12⟹x=6$

substitute x=0 we get, $3y=-12⟹y=-4$

therefore, $3y-2x=-12$ line passes through (6,0) and (0,-4) as shown in fig.

Hence, $3y-2x\ge -12$includes the region above the line.

similarly for $2x+5y=20$

substitute y=0 we get, $2x=20⟹x=10$

substitute x=0 we get, $5y=20⟹y=4$

therefore, $2x+5y=20$ line passes through (10,0) and (0,4) as shown in fig.

Hence, $2x+5y\le 20$includes the region below the line

the shaded region as shown in figure is intersection region

distance between (0,4) and (0,-4) is 8 units

adding $3y-2x=-12$ and $2x+5y=20⟹y=1$

substituting y=1 in any one of equation gives $x=7.5$

therefore, shaded triangle base =8 units and

height = 7.5 unit

Area = $\frac{1}{2}\times base\times height$

$=\frac{1}{2}\times 8\times 7.5=30$