Rectangular Plot Width Afd0A1

1. Problem 1: Find the possible widths of a rectangular plot where the area is at least 60 m² and the length is 4 meters longer than the width. 2. Let the width be $w$ meters. Then the length is $w + 4$ meters. 3. The area $A$ of a rectangle is given by the formula: $$A = \text{length} \times \text{width}$$ 4. Substitute the expressions for length and width: $$A = w(w + 4)$$ 5. The area must be at least 60 m², so: $$w(w + 4) \geq 60$$ 6. Expand the left side: $$w^2 + 4w \geq 60$$ 7. Bring all terms to one side to form a quadratic inequality: $$w^2 + 4w - 60 \geq 0$$ 8. Solve the quadratic equation $w^2 + 4w - 60 = 0$ to find critical points. 9. Use the quadratic formula: $$w = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ where $a=1$, $b=4$, and $c=-60$. 10. Calculate the discriminant: $$\Delta = 4^2 - 4 \times 1 \times (-60) = 16 + 240 = 256$$ 11. Calculate the roots: $$w = \frac{-4 \pm \sqrt{256}}{2} = \frac{-4 \pm 16}{2}$$ 12. Roots are: $$w_1 = \frac{-4 + 16}{2} = \frac{12}{2} = 6$$ $$w_2 = \frac{-4 - 16}{2} = \frac{-20}{2} = -10$$ 13. Since width cannot be negative, discard $w = -10$. 14. The quadratic inequality $w^2 + 4w - 60 \geq 0$ is satisfied when $w \leq -10$ or $w \geq 6$. 15. Only $w \geq 6$ is valid for width. 16. Therefore, the possible widths are: $$\boxed{w \geq 6 \text{ meters}}$$ --- 17. Problem 2: Find the time interval when the stone's height $h$ is greater than 20 meters. 18. Given height function: $$h = -5t^2 + 25t$$ 19. We want to find $t$ such that: $$-5t^2 + 25t > 20$$ 20. Bring all terms to one side: $$-5t^2 + 25t - 20 > 0$$ 21. Multiply both sides by $-1$ to simplify, remembering to reverse inequality: $$5t^2 - 25t + 20 < 0$$ 22. Divide entire inequality by 5: $$t^2 - 5t + 4 < 0$$ 23. Factor the quadratic: $$t^2 - 5t + 4 = (t - 4)(t - 1)$$ 24. The inequality $(t - 4)(t - 1) < 0$ holds when $t$ is between the roots: $$1 < t < 4$$ 25. Therefore, the stone is higher than 20 meters during the time interval: $$\boxed{(1, 4) \text{ seconds}}$$