1. **State the problem:** We want to maximize the objective function $$P = 22x_1 + 11x_2$$ subject to the constraints: $$2x_1 + 9x_2 \leq 117$$ $$x_1 + 4x_2 \leq 53$$ $$x_1, x_2 \geq 0$$ 2. **Identify the feasible region:** The feasible region is bounded by the inequalities above and the non-negativity constraints. 3. **Find the corner points of the feasible region:** - Intersection with axes: - When $$x_1=0$$, from $$2(0) + 9x_2 \leq 117$$, $$x_2 \leq 13$$. - When $$x_2=0$$, from $$2x_1 + 9(0) \leq 117$$, $$x_1 \leq 58.5$$. - When $$x_1=0$$, from $$0 + 4x_2 \leq 53$$, $$x_2 \leq 13.25$$. - When $$x_2=0$$, from $$x_1 + 0 \leq 53$$, $$x_1 \leq 53$$. 4. **Find intersection of the two constraints:** Solve the system: $$\begin{cases} 2x_1 + 9x_2 = 117 \\ x_1 + 4x_2 = 53 \end{cases}$$ Multiply second equation by 2: $$2x_1 + 8x_2 = 106$$ Subtract from first: $$2x_1 + 9x_2 - (2x_1 + 8x_2) = 117 - 106$$ $$x_2 = 11$$ Substitute back: $$x_1 + 4(11) = 53 \Rightarrow x_1 + 44 = 53 \Rightarrow x_1 = 9$$ 5. **Evaluate P at corner points:** - At $$x_1=0, x_2=0$$: $$P=0$$ - At $$x_1=0, x_2=13$$ (from first constraint): $$P=22(0)+11(13)=143$$ - At $$x_1=53, x_2=0$$ (from second constraint): $$P=22(53)+11(0)=1166$$ - At intersection $$x_1=9, x_2=11$$: $$P=22(9)+11(11)=198+121=319$$ 6. **Check feasibility of points:** - $$x_1=53, x_2=0$$ satisfies both constraints. - $$x_1=0, x_2=13$$ satisfies both constraints. - Intersection point $$x_1=9, x_2=11$$ satisfies both constraints. 7. **Determine maximum:** The maximum value of $$P$$ is $$1166$$ at $$x_1=53, x_2=0$$. **Final answer:** $$\boxed{P_{max} = 1166 \text{ at } (x_1, x_2) = (53, 0)}$$