1. **State the problem:** Jacob needs to buy meat and cheese to meet minimum weekly requirements of 12 units of carbohydrates and 8 units of protein. 2. **Define variables:** Let $x$ = pounds of meat, $y$ = pounds of cheese. 3. **Write constraints based on nutrients:** - Carbohydrates: $7x + 3y \geq 12$ - Protein: $2x + y \geq 8$ - Non-negativity: $x \geq 0$, $y \geq 0$ 4. **Cost function to minimize:** $$C = 3.5x + 2.7y$$ 5. **Solve constraints as equalities to find corner points:** From $7x + 3y = 12$, express $y = \frac{12 - 7x}{3}$. From $2x + y = 8$, express $y = 8 - 2x$. 6. **Find intersection of constraints:** Set $\frac{12 - 7x}{3} = 8 - 2x$: $$12 - 7x = 24 - 6x$$ $$-7x + 6x = 24 - 12$$ $$-x = 12$$ $$x = -12$$ (not feasible since $x \geq 0$) 7. **Check intercepts:** - For $7x + 3y = 12$: - If $x=0$, $y=4$ - If $y=0$, $x=\frac{12}{7} \approx 1.71$ - For $2x + y = 8$: - If $x=0$, $y=8$ - If $y=0$, $x=4$ 8. **Check feasible corner points:** - Point A: $(0,8)$ satisfies both constraints. - Point B: $(1.71,0)$ check protein: $2(1.71)+0=3.42<8$ no. - Point C: $(0,4)$ check protein: $2(0)+4=4<8$ no. - Point D: $(4,0)$ check carbs: $7(4)+3(0)=28>12$ yes, protein: $2(4)+0=8$ yes. 9. **Evaluate cost at feasible points:** - At A $(0,8)$: $C=3.5(0)+2.7(8)=21.6$ - At D $(4,0)$: $C=3.5(4)+2.7(0)=14$ 10. **Check intersection of constraints for feasibility:** Since intersection $x=-12$ is not feasible, check boundary lines. 11. **Check point where $7x+3y=12$ and $2x+y=8$ meet in feasible region:** Try to find feasible point on $7x+3y=12$ with $2x+y \geq 8$. Try $x=1$, then $y=\frac{12-7}{3}=\frac{5}{3} \approx 1.67$. Check protein: $2(1)+1.67=3.67<8$ no. Try $x=2$, $y=\frac{12-14}{3}=-\frac{2}{3}$ no. Try $x=0$, $y=4$ protein=4 no. Try $x=3$, $y=\frac{12-21}{3}=-3$ no. 12. **Check point where $2x + y = 8$ and $7x + 3y \geq 12$:** Express $y=8-2x$. Substitute into $7x + 3y \geq 12$: $$7x + 3(8 - 2x) \geq 12$$ $$7x + 24 - 6x \geq 12$$ $$x + 24 \geq 12$$ $$x \geq -12$$ always true for $x \geq 0$. So the feasible region is bounded by $2x + y = 8$ and $x,y \geq 0$ with $7x + 3y \geq 12$. 13. **Find point on $2x + y = 8$ that satisfies $7x + 3y = 12$:** Substitute $y=8-2x$ into $7x + 3y = 12$: $$7x + 3(8 - 2x) = 12$$ $$7x + 24 - 6x = 12$$ $$x + 24 = 12$$ $$x = -12$$ no. 14. **Since no intersection in positive quadrant, check corner points:** - $(0,8)$ cost = 21.6 - $(4,0)$ cost = 14 - Check if $(1.5,5) satisfies constraints: Carbs: $7(1.5)+3(5)=10.5+15=25.5 \geq 12$ yes Protein: $2(1.5)+5=3+5=8 \geq 8$ yes Cost: $3.5(1.5)+2.7(5)=5.25+13.5=18.75$ 15. **Minimum cost is at $(4,0)$ pounds:** Jacob should buy 4 pounds of meat and 0 pounds of cheese. 16. **Minimum cost:** $$C = 3.5 \times 4 + 2.7 \times 0 = 14$$ **Final answer:** Jacob should buy **4** pounds of meat and **0** pounds of cheese to minimize cost. The minimum cost is **14**.