1. **Problem statement:** A tree broken by the wind forms a right-angled triangle with the ground. The broken part of the tree makes an angle of 60° with the ground. The top of the tree is 15 m far from its foot. (i) If the length of the broken part is $x$ m, calculate the value of $x$. (ii) How tall was the tree originally? (iii) Compare the distance and length of the broken part of the tree. 2. **Formula and rules:** - We use trigonometric ratios in right triangles. Here, the broken part of the tree is the hypotenuse, the distance from the foot to the top is the adjacent side to the 60° angle. - Cosine of an angle in a right triangle is defined as: $$\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}}$$ 3. **Step-by-step solution:** (i) Calculate $x$: - Given angle $\theta = 60^\circ$, adjacent side = 15 m, hypotenuse = $x$. - Using cosine: $$\cos 60^\circ = \frac{15}{x}$$ - We know $\cos 60^\circ = 0.5$, so: $$0.5 = \frac{15}{x}$$ - Multiply both sides by $x$: $$0.5x = 15$$ - Divide both sides by 0.5: $$x = \frac{15}{0.5} = 30$$ - So, the length of the broken part is $x = 30$ m. (ii) Find the original height of the tree: - The original height is the sum of the unbroken part (vertical leg) plus the vertical component of the broken part. - The vertical component of the broken part is: $$x \sin 60^\circ = 30 \times \frac{\sqrt{3}}{2} = 15\sqrt{3}$$ - The horizontal distance is 15 m, so the unbroken part is the vertical leg of the right triangle formed by the tree's base and the broken part. - Since the broken part forms a right triangle with the ground, the unbroken part is the vertical leg adjacent to the right angle. - The unbroken part is: $$\sqrt{x^2 - 15^2} = \sqrt{30^2 - 15^2} = \sqrt{900 - 225} = \sqrt{675} = 15\sqrt{3}$$ - Therefore, the total height of the tree is: $$15\sqrt{3} + 15\sqrt{3} = 30\sqrt{3} \approx 51.96$$ m. (iii) Compare the distance and length of the broken part: - Distance from foot to top = 15 m. - Length of broken part = 30 m. - The broken part is twice as long as the horizontal distance from the foot to the top. **Final answers:** (i) $x = 30$ m (ii) Original height of the tree $= 30\sqrt{3} \approx 51.96$ m (iii) The broken part is twice the horizontal distance from the foot to the top.