A surveyor wants to know the length of a tunnel built through a mountain. According to his equipment, he is located 212 meters from one entrance of the tunnel, at an angle of $57^{\circ}$ to the perpendicular. Also according to his equipment, he is 119 meters from the other entrance of the tunnel, at an angle of $14^{\circ}$ to the perpendicular. Based on these measurements, find the length of the entire tunnel.
Do not round any intermediate computations. Round your answer to the nearest tenth.
Note that the figure below is not drawn to scale.
Answer
Round the length of the tunnel to the nearest tenth. The length of the tunnel is approximately \(\boxed{230.9}\) meters.
Step 1 ：Given that the surveyor is located 212 meters from one entrance of the tunnel, at an angle of \(57^{\circ}\) to the perpendicular, and 119 meters from the other entrance of the tunnel, at an angle of \(14^{\circ}\) to the perpendicular.
Step 2 ：Convert the angles from degrees to radians. For \(57^{\circ}\), it is approximately 0.9948376736367679 radians. For \(14^{\circ}\), it is approximately 0.24434609527920614 radians.
Step 3 ：Calculate the lengths of the sides of the triangles adjacent to the angles using the cosine of the angles and the lengths of the hypotenuses of the triangles. For the first triangle, the length is approximately 115.46347542318574 meters. For the second triangle, the length is approximately 115.46519142684357 meters.
Step 4 ：Add the lengths of the two sides to find the length of the entire tunnel. The length of the tunnel is approximately 230.92866685002932 meters.
Step 5 ：Round the length of the tunnel to the nearest tenth. The length of the tunnel is approximately \(\boxed{230.9}\) meters.
