Question

a surveyor wants to know the length of a tunnel built through a mountain. according to her equipment, she is located 54 meters from one entrance of the tunnel, at an angle of 56° to the perpendicular. also according to her equipment, she is 31 meters from the other entrance of the tunnel, at an angle of 13° 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. 54 meters 56° 13° 31 meters

Explanation:

Step1: Use the cosine function for each right - triangle

For the first right - triangle with hypotenuse \(h_1 = 54\) meters and angle \(\theta_1=56^{\circ}\), the adjacent side \(x_1\) (part of the perpendicular) is given by \(x_1 = 54\cos(56^{\circ})\).
For the second right - triangle with hypotenuse \(h_2 = 31\) meters and angle \(\theta_2 = 13^{\circ}\), the adjacent side \(x_2\) (part of the perpendicular) is given by \(x_2=31\cos(13^{\circ})\).

Step2: Calculate \(x_1\) and \(x_2\)

We know that \(\cos(56^{\circ})\approx0.5592\) and \(\cos(13^{\circ})\approx0.9744\).
So, \(x_1 = 54\times0.5592=30.1968\) and \(x_2 = 31\times0.9744 = 30.2064\).

Step3: Find the length of the tunnel

The length of the tunnel \(L=x_1 + x_2\).
\(L=30.1968+30.2064=60.4032\approx60.4\)

Answer:

\(60.4\)