1. **Problem 3:** A tree is supported by a guy wire anchored 7.0 m from the base of the tree. The angle between the wire and the ground is 60°. Find how far up the tree the wire reaches. 2. **Formula:** Use the sine function in a right triangle: $$\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}$$ where the opposite side is the height up the tree, and the hypotenuse is the length of the wire. 3. **Given:** - Distance from base (adjacent side) = 7.0 m - Angle with ground $$\theta = 60^\circ$$ 4. **Find the hypotenuse (length of wire):** Use cosine since adjacent and hypotenuse are known: $$\cos(60^\circ) = \frac{7.0}{\text{hypotenuse}}$$ 5. Solve for hypotenuse: $$\text{hypotenuse} = \frac{7.0}{\cos(60^\circ)}$$ $$= \frac{7.0}{0.5} = 14.0\text{ m}$$ 6. **Find height up the tree (opposite side):** $$\sin(60^\circ) = \frac{\text{height}}{14.0}$$ 7. Solve for height: $$\text{height} = 14.0 \times \sin(60^\circ)$$ $$= 14.0 \times 0.8660 = 12.1\text{ m}$$ --- 8. **Problem 4:** A flagpole is 14.0 m high. The angle of elevation from Jon's position to the top is 63°. Find Jon's distance from the flagpole. 9. **Formula:** Use tangent function: $$\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}$$ 10. **Given:** - Height (opposite) = 14.0 m - Angle $$\theta = 63^\circ$$ 11. Solve for adjacent (distance): $$\tan(63^\circ) = \frac{14.0}{\text{distance}}$$ 12. Rearrange: $$\text{distance} = \frac{14.0}{\tan(63^\circ)}$$ 13. Calculate: $$\tan(63^\circ) \approx 1.9626$$ $$\text{distance} = \frac{14.0}{1.9626} = 7.1\text{ m}$$ **Final answers:** - Problem 3: The wire reaches approximately **12.1 m** up the tree. - Problem 4: Jon is approximately **7.1 m** from the flagpole.