Discussion
Which one of the options given represents the feasible region of the linear programming model:
\(\mathit{Maximize}\ 45X_{1} + 60X_{2}\)
\(X_{1} \le 45\)
\(X_{2} \le 50\)
\(10X_{1} + 10X_{2} \ge 600\)
\(25X_{1} + 5X_{2} \le 750\)
Answer: B
The feasible region is Region Q.
Simplify the constraints: \(10X_{1} + 10X_{2} \ge 600\) is \(X_{1} + X_{2} \ge 60\), and \(25X_{1} + 5X_{2} \le 750\) is \(5X_{1} + X_{2} \le 150\). A feasible point must lie
- below the horizontal line \(X_{2} = 50\);
- on or above the gently sloping line \(X_{1} + X_{2} = 60\) (intercept 60 on each axis);
- on or left of the steep line \(5X_{1} + X_{2} = 150\) (intercept 30 on the \(X_{1}\) axis), which already keeps \(X_{1} \le 30 \lt 45\).
These meet in a small triangle with corners (10, 50), (20, 50) and (22.5, 37.5), the shaded Region Q.
Region P lies below \(X_{1} + X_{2} = 60\), so it breaks the \(\ge 600\) constraint. Region S lies to the right of the steep line, breaking \(25X_{1} + 5X_{2} \le 750\). Region R breaks both.
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