Question

a surveyor wants to know the length of a tunnel built through a mountain. according to her equipment, she is located 325 meters from one entrance of the tunnel, at an angle of 48° to the perpendicular. also according to her equipment, she is 230 meters from the other entrance of the tunnel, at an angle of 19° 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.

Explanation:

Step1: Use cosine - law formula

Let \(a = 325\), \(b = 230\), and \(\theta=48^{\circ}+19^{\circ}=67^{\circ}\). The cosine - law is \(c^{2}=a^{2}+b^{2}-2ab\cos\theta\).

Step2: Substitute values

\(c^{2}=325^{2}+230^{2}-2\times325\times230\times\cos(67^{\circ})\).
\(c^{2}=105625 + 52900-149500\times0.39073\).
\(c^{2}=105625+52900 - 58414.135\).
\(c^{2}=100110.865\).

Step3: Find \(c\)

\(c=\sqrt{100110.865}\approx316.4\).

Answer:

\(316.4\)