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 21° 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: Find the two segments of the tunnel

Let the distance from the surveyor to the tunnel (the perpendicular distance) be \( h \). For the first triangle (with hypotenuse 340 meters and angle \( 58^\circ \)):
\( \cos(58^\circ)=\frac{\text{adjacent}}{\text{hypotenuse}}=\frac{x}{340} \), so \( x = 340\cos(58^\circ) \)
For the second triangle (with hypotenuse 193 meters and angle \( 21^\circ \)):
\( \cos(21^\circ)=\frac{\text{adjacent}}{\text{hypotenuse}}=\frac{y}{193} \), so \( y = 193\cos(21^\circ) \)

Step2: Calculate the length of the tunnel

The length of the tunnel \( L=x + y \)
First, calculate \( x = 340\cos(58^\circ)\approx340\times0.5299\approx180.166 \)
Then, calculate \( y = 193\cos(21^\circ)\approx193\times0.9336\approx179.1848 \)
Now, \( L=180.166 + 179.1848\approx359.3508 \)

Answer:

\( 359.4 \) (rounded to the nearest tenth)