1. **State the problem:** We want to maximize the objective function $$P = 20x + 80y$$ subject to the constraints $$x + 3y \leq 12$$ and $$x, y \geq 0$$. 2. **Identify the feasible region:** The constraints define a triangular feasible region bounded by the axes and the line $$x + 3y = 12$$. 3. **Find the corner points of the feasible region:** - When $$x=0$$, $$3y=12 \Rightarrow y=4$$. - When $$y=0$$, $$x=12$$. - The third corner is at the origin $$(0,0)$$. 4. **Evaluate the objective function at each corner:** - At $$(0,0)$$: $$P=20(0)+80(0)=0$$. - At $$(12,0)$$: $$P=20(12)+80(0)=240$$. - At $$(0,4)$$: $$P=20(0)+80(4)=320$$. 5. **Determine the maximum value:** The maximum value of $$P$$ is $$320$$ at $$(0,4)$$. **Final answer:** $$\boxed{320}$$