1. **State the problem:** We want to maximize the objective function $$P = 35x_1 + 140x_2$$ subject to the constraint $$x_1 + 5x_2 \leq 20$$ and non-negativity constraints $$x_1, x_2 \geq 0$$. 2. **Understand the constraints and feasible region:** The inequality $$x_1 + 5x_2 \leq 20$$ describes a line and the area below it including the boundary. Since $$x_1, x_2 \geq 0$$, the feasible region is bounded by the axes and the line. 3. **Find the intercepts of the constraint line:** - When $$x_1 = 0$$, $$5x_2 = 20 \Rightarrow x_2 = 4$$. - When $$x_2 = 0$$, $$x_1 = 20$$. 4. **Evaluate the objective function at the vertices of the feasible region:** Vertices are at points where constraints intersect: - Point A: $$(0,0)$$ - Point B: $$(20,0)$$ - Point C: $$(0,4)$$ Calculate $$P$$ at each vertex: - At A: $$P = 35(0) + 140(0) = 0$$ - At B: $$P = 35(20) + 140(0) = 700$$ - At C: $$P = 35(0) + 140(4) = 560$$ 5. **Determine the maximum value:** The maximum $$P$$ is $$700$$ at $$x_1 = 20$$ and $$x_2 = 0$$. 6. **Answer:** A. The optimal solution is $$P = 700$$ when $$x_1 = 20$$ and $$x_2 = 0$$.