1. **State the problem:** We have a rectangle with length 35 feet and width $w$ feet. The area must be at least 420 square feet, and the perimeter must be no more than 125 feet. 2. **Part (a): Could $w=30$ feet?** - Area check: $35 \times 30 = 1050$ which is greater than 420, so area condition is satisfied. - Perimeter check: $2 \times (35 + 30) = 2 \times 65 = 130$ which is greater than 125, so perimeter condition is NOT satisfied. - **Answer:** No, $w=30$ feet is not possible because the perimeter exceeds 125 feet. 3. **Part (b): Area inequality and solution** - Area formula: $\text{Area} = 35 \times w$ - Inequality: $$35w \geq 420$$ - Solve for $w$: $$w \geq \frac{420}{35}$$ $$w \geq 12$$ 4. **Part (c): Perimeter inequality and solution** - Perimeter formula: $$P = 2(35 + w)$$ - Inequality: $$2(35 + w) \leq 125$$ - Divide both sides by 2: $$\cancel{2}(35 + w) \leq \frac{125}{\cancel{2}}$$ $$35 + w \leq 62.5$$ - Solve for $w$: $$w \leq 62.5 - 35$$ $$w \leq 27.5$$ 5. **Part (d): Values of $w$ satisfying both inequalities** - From (b): $w \geq 12$ - From (c): $w \leq 27.5$ - Combined solution: $$12 \leq w \leq 27.5$$ - Example values: $w=15$, $w=20$, $w=27$ all satisfy both conditions. **Final answers:** - (a) No, $w=30$ is not possible because perimeter exceeds 125 feet. - (b) $w \geq 12$ - (c) $w \leq 27.5$ - (d) $12 \leq w \leq 27.5$