1. **State the problem:** We want to maximize the profit function $$\text{Profit} = 5000x + 7000y$$ subject to constraints: $$4 \leq x \leq 24$$ $$x + y \leq 24$$ $$8x + 7y \leq 230$$ where $$x$$ and $$y$$ represent hours or units related to silage and hay production respectively. 2. **Rewrite the constraints to find bounds on $$y$$:** From $$x + y \leq 24$$, we have $$y \leq 24 - x$$. From $$8x + 7y \leq 230$$, rearranged: $$7y \leq 230 - 8x$$ $$y \leq \frac{230 - 8x}{7}$$. 3. **Determine feasible region:** - $$x$$ ranges between 4 and 24. - $$y$$ is limited by the minimum of $$24 - x$$ and $$\frac{230 - 8x}{7}$$, and must be nonnegative (implied in context). 4. **Find corner points of feasible region:** Check intersections and boundary values for $$x$$ and $$y$$: - At $$x=4$$: $$y \leq 24-4=20$$ and $$y \leq \frac{230-32}{7} = \frac{198}{7} \approx 28.29$$ So $$y_{max}=20$$ Point: (4,20) - At $$x=24$$: $$y \leq 24-24=0$$ and $$y \leq \frac{230-192}{7} = \frac{38}{7} \approx 5.43$$ So $$y_{max}=0$$ Point: (24,0) - Intersection of $$x + y = 24$$ and $$8x + 7y = 230$$: Solve: $$y = 24 - x$$ Substitute into second: $$8x + 7(24 - x) = 230$$ $$8x + 168 - 7x = 230$$ $$x + 168 = 230$$ $$x = 62$$ (not in acceptable range since max $$x=24$$) So no intersection within bounds. - Check intersection at $$x=4$$ or $$x=24$$ already done. - Check $$y=0$$ line: For $$y=0$$, check constraints: $$4 \leq x \leq 24$$ and $$8x \leq 230$$ $$x \leq \frac{230}{8} = 28.75$$ (allowed) So points at (4,0) and (24,0) are feasible. 5. **Evaluate profit at corner points:** - At (4, 20): $$5000 \times 4 + 7000 \times 20 = 20000 + 140000 = 160000$$ - At (24, 0): $$5000 \times 24 + 7000 \times 0 = 120000 + 0 = 120000$$ - At (4, 0): $$5000 \times 4 + 7000 \times 0 = 20000$$ 6. **Identify the maximum profit:** Maximum profit is $$160000$$ at $$x=4$$ and $$y=20$$. **Final answer:** The maximum profit is **160000** obtained when $$x=4$$ hours and $$y=20$$ units.