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 58° to the perpendicular. also according to his equipment, he is 193 meters from the other entrance of the tunnel, at an angle of 31° 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: Calculate the length from the first part

We use the cosine function. For the 340 - meter side with an angle of \(58^{\circ}\) to the perpendicular. The adjacent side (let's call it \(x_1\)) to the angle in a right - triangle (where the surveyor's line is the hypotenuse) is given by \(x_1 = 340\cos(58^{\circ})\).
Using a calculator, \(\cos(58^{\circ})\approx0.5299\), so \(x_1 = 340\times0.5299= 180.166\) meters.

Step2: Calculate the length from the second part

For the 193 - meter side with an angle of \(31^{\circ}\) to the perpendicular. The adjacent side (let's call it \(x_2\)) to the angle in a right - triangle (where the surveyor's line is the hypotenuse) is given by \(x_2=193\cos(31^{\circ})\).
Using a calculator, \(\cos(31^{\circ})\approx0.8572\), so \(x_2 = 193\times0.8572 = 165.4396\) meters.

Step3: Calculate the total length of the tunnel

The total length \(L\) of the tunnel is \(L=x_1 + x_2\).
\(L=180.166+165.4396=345.6056\) meters.

Answer:

\(345.6\) meters