Question

a surveyor wants to know the length of a tunnel built through a mountain. according to his equipment, he is located 340 meters from one entrance of the tunnel, at an angle of 21° to the perpendicular. also according to his equipment, he is 193 meters from the other entrance of the tunnel, at an angle of 58° 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 concept

Let the two distances from surveyor to tunnel entrances be $a = 340$ and $b = 193$, and the included - angle $\theta=58^{\circ}+21^{\circ}=79^{\circ}$. The length of the tunnel $d$ can be found using the law of cosines $d^{2}=a^{2}+b^{2}-2ab\cos\theta$.

Step2: Substitute values

$d^{2}=340^{2}+193^{2}-2\times340\times193\times\cos(79^{\circ})$
$d^{2}=115600 + 37249-132040\times0.1908$
$d^{2}=115600 + 37249-25193.232$
$d^{2}=127655.768$
$d=\sqrt{127655.768}\approx357.3$

Answer:

$357.3$