1. **State the problem:** We need to find the height of a balloon observed from two stations X and Y, which are 3000 feet apart. Given angles are horizontal angles and angle of elevation. 2. **Given data:** - Distance between X and Y: $3000$ ft - At X: horizontal angle between balloon and C is $75^\circ 25'$ (convert to decimal: $75 + \frac{25}{60} = 75.4167^\circ$) - Angle of elevation at X: $18^\circ$ - At Y: horizontal angle between balloon and X is $64^\circ 30'$ (convert to decimal: $64 + \frac{30}{60} = 64.5^\circ$) 3. **Approach:** - Use the horizontal angles to find the horizontal distance from X to the balloon. - Use the angle of elevation and horizontal distance to find the height. 4. **Calculate the horizontal distance from X to the balloon:** - The sum of horizontal angles at X and Y is $75.4167^\circ + 64.5^\circ = 139.9167^\circ$ - The angle at balloon between X and Y is $180^\circ - 139.9167^\circ = 40.0833^\circ$ 5. **Use the Law of Sines in triangle X-Y-Balloon:** $$\frac{XY}{\sin(40.0833^\circ)} = \frac{XB}{\sin(64.5^\circ)} = \frac{YB}{\sin(75.4167^\circ)}$$ 6. **Calculate $XB$ (distance from X to balloon):** $$XB = \frac{XY \times \sin(64.5^\circ)}{\sin(40.0833^\circ)} = \frac{3000 \times \sin(64.5^\circ)}{\sin(40.0833^\circ)}$$ Calculate the sines: - $\sin(64.5^\circ) \approx 0.9004$ - $\sin(40.0833^\circ) \approx 0.6441$ So, $$XB = \frac{3000 \times 0.9004}{0.6441} \approx 4191.5 \text{ feet}$$ 7. **Calculate the height of the balloon using angle of elevation at X:** $$\text{height} = XB \times \tan(18^\circ)$$ Calculate $\tan(18^\circ) \approx 0.3249$ So, $$\text{height} = 4191.5 \times 0.3249 \approx 1361.3 \text{ feet}$$ **Final answer:** The height of the balloon is approximately $1361.3$ feet.