Let's find the corner points (vertices) of the bounded feasible region defined by the inequalities:
1) $5x + y ≤ 100<br>2)x + y ≤ 60<br>3)x ≥ 0,y ≥ 0\displaystyle <br><br>First, let's find the intercepts of the boundary lines on the coordinate axes:<br>- For the line $5x + y = 100: intercepts are (20,0) and (0,100).
- For the line x+y=60: intercepts are (60,0) and (0,60).
Next, we find the intersection point of $5x + y = 100andx + y = 60:<br>Subtractingthesecondequationfromthefirst:<br><spanclass="katex−display"><spanclass="katex"><spanclass="katex−mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"display="block"><semantics><mrow><mostretchy="false">(</mo><mn>5</mn><mi>x</mi><mo>+</mo><mi>y</mi><mostretchy="false">)</mo><mo>−</mo><mostretchy="false">(</mo><mi>x</mi><mo>+</mo><mi>y</mi><mostretchy="false">)</mo><mo>=</mo><mn>100</mn><mo>−</mo><mn>60</mn><mtext> </mtext><mo>⟹</mo><mtext> </mtext><mn>4</mn><mi>x</mi><mo>=</mo><mn>40</mn><mtext> </mtext><mo>⟹</mo><mtext> </mtext><mi>x</mi><mo>=</mo><mn>10</mn></mrow><annotationencoding="application/x−tex">(5x+y)−(x+y)=100−60⟹4x=40⟹x=10</annotation></semantics></math></span><spanclass="katex−html"aria−hidden="true"><spanclass="base"><spanclass="strut"style="height:1em;vertical−align:−0.25em;"></span><spanclass="mopen">(</span><spanclass="mord">5</span><spanclass="mordmathnormal">x</span><spanclass="mspace"style="margin−right:0.2222em;"></span><spanclass="mbin">+</span><spanclass="mspace"style="margin−right:0.2222em;"></span></span><spanclass="base"><spanclass="strut"style="height:1em;vertical−align:−0.25em;"></span><spanclass="mordmathnormal"style="margin−right:0.03588em;">y</span><spanclass="mclose">)</span><spanclass="mspace"style="margin−right:0.2222em;"></span><spanclass="mbin">−</span><spanclass="mspace"style="margin−right:0.2222em;"></span></span><spanclass="base"><spanclass="strut"style="height:1em;vertical−align:−0.25em;"></span><spanclass="mopen">(</span><spanclass="mordmathnormal">x</span><spanclass="mspace"style="margin−right:0.2222em;"></span><spanclass="mbin">+</span><spanclass="mspace"style="margin−right:0.2222em;"></span></span><spanclass="base"><spanclass="strut"style="height:1em;vertical−align:−0.25em;"></span><spanclass="mordmathnormal"style="margin−right:0.03588em;">y</span><spanclass="mclose">)</span><spanclass="mspace"style="margin−right:0.2778em;"></span><spanclass="mrel">=</span><spanclass="mspace"style="margin−right:0.2778em;"></span></span><spanclass="base"><spanclass="strut"style="height:0.7278em;vertical−align:−0.0833em;"></span><spanclass="mord">100</span><spanclass="mspace"style="margin−right:0.2222em;"></span><spanclass="mbin">−</span><spanclass="mspace"style="margin−right:0.2222em;"></span></span><spanclass="base"><spanclass="strut"style="height:0.6684em;vertical−align:−0.024em;"></span><spanclass="mord">60</span><spanclass="mspace"style="margin−right:0.2778em;"></span><spanclass="mspace"style="margin−right:0.2778em;"></span><spanclass="mrel">⟹</span><spanclass="mspace"style="margin−right:0.2778em;"></span><spanclass="mspace"style="margin−right:0.2778em;"></span></span><spanclass="base"><spanclass="strut"style="height:0.6444em;"></span><spanclass="mord">4</span><spanclass="mordmathnormal">x</span><spanclass="mspace"style="margin−right:0.2778em;"></span><spanclass="mrel">=</span><spanclass="mspace"style="margin−right:0.2778em;"></span></span><spanclass="base"><spanclass="strut"style="height:0.6684em;vertical−align:−0.024em;"></span><spanclass="mord">40</span><spanclass="mspace"style="margin−right:0.2778em;"></span><spanclass="mspace"style="margin−right:0.2778em;"></span><spanclass="mrel">⟹</span><spanclass="mspace"style="margin−right:0.2778em;"></span><spanclass="mspace"style="margin−right:0.2778em;"></span></span><spanclass="base"><spanclass="strut"style="height:0.4306em;"></span><spanclass="mordmathnormal">x</span><spanclass="mspace"style="margin−right:0.2778em;"></span><spanclass="mrel">=</span><spanclass="mspace"style="margin−right:0.2778em;"></span></span><spanclass="base"><spanclass="strut"style="height:0.6444em;"></span><spanclass="mord">10</span></span></span></span></span><br>Substitutingx = 10intox + y = 60:<br><spanclass="katex−display"><spanclass="katex"><spanclass="katex−mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"display="block"><semantics><mrow><mn>10</mn><mo>+</mo><mi>y</mi><mo>=</mo><mn>60</mn><mtext> </mtext><mo>⟹</mo><mtext> </mtext><mi>y</mi><mo>=</mo><mn>50</mn></mrow><annotationencoding="application/x−tex">10+y=60⟹y=50</annotation></semantics></math></span><spanclass="katex−html"aria−hidden="true"><spanclass="base"><spanclass="strut"style="height:0.7278em;vertical−align:−0.0833em;"></span><spanclass="mord">10</span><spanclass="mspace"style="margin−right:0.2222em;"></span><spanclass="mbin">+</span><spanclass="mspace"style="margin−right:0.2222em;"></span></span><spanclass="base"><spanclass="strut"style="height:0.625em;vertical−align:−0.1944em;"></span><spanclass="mordmathnormal"style="margin−right:0.03588em;">y</span><spanclass="mspace"style="margin−right:0.2778em;"></span><spanclass="mrel">=</span><spanclass="mspace"style="margin−right:0.2778em;"></span></span><spanclass="base"><spanclass="strut"style="height:0.6684em;vertical−align:−0.024em;"></span><spanclass="mord">60</span><spanclass="mspace"style="margin−right:0.2778em;"></span><spanclass="mspace"style="margin−right:0.2778em;"></span><spanclass="mrel">⟹</span><spanclass="mspace"style="margin−right:0.2778em;"></span><spanclass="mspace"style="margin−right:0.2778em;"></span></span><spanclass="base"><spanclass="strut"style="height:0.625em;vertical−align:−0.1944em;"></span><spanclass="mordmathnormal"style="margin−right:0.03588em;">y</span><spanclass="mspace"style="margin−right:0.2778em;"></span><spanclass="mrel">=</span><spanclass="mspace"style="margin−right:0.2778em;"></span></span><spanclass="base"><spanclass="strut"style="height:0.6444em;"></span><spanclass="mord">50</span></span></span></span></span><br>Sotheintersectionpointis(10, 50).<br><br>Theboundarypointsdefiningthefeasibleregioninthefirstquadrantare:<br>−Origin:(0, 0)<br>−x−axisboundary:(20, 0)$ (since it lies inside $x + y ≤ 60)<br>−Intersectionpoint:(10, 50)<br>−y−axisboundary:(0, 60)≤ …" style="color:#cc0000">\displaystyle (since it lies inside $5x + y ≤ 100)
Therefore, the corner points are
(0,0),
(20,0),
(10,50), and
(0,60).
This corresponds to Option A.
Hence, **Option A** is the correct answer.