1. **State the problem:** We want to maximize the objective function $$P = 10x + 70y$$ subject to the constraint $$x + 5y \leq 20$$ and the nonnegativity conditions $$x \geq 0, y \geq 0$$. 2. **Understand the constraints:** The feasible region is defined by $$x + 5y \leq 20$$, $$x \geq 0$$, and $$y \geq 0$$. This region is a triangle bounded by the axes and the line $$x + 5y = 20$$. 3. **Find the corner points of the feasible region:** - When $$x=0$$, $$5y=20 \Rightarrow y=4$$. - When $$y=0$$, $$x=20$$. - The third corner is at the origin $$(0,0)$$. 4. **Evaluate the objective function at each corner:** - At $$(0,0)$$: $$P=10(0)+70(0)=0$$ - At $$(20,0)$$: $$P=10(20)+70(0)=200$$ - At $$(0,4)$$: $$P=10(0)+70(4)=280$$ 5. **Determine the maximum value:** The maximum value of $$P$$ is $$280$$ at $$(0,4)$$. **Final answer:** $$\boxed{280}$$