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 segment from 340m side

We have a right triangle with hypotenuse 340 m and angle \( 58^\circ \). Let the length of the tunnel segment from the first entrance be \( x \). Using the cosine function (\( \cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}} \)):
\( \cos(58^\circ)=\frac{x}{340} \)
\( x = 340\times\cos(58^\circ) \)
Calculate \( \cos(58^\circ)\approx0.5299 \), so \( x\approx340\times0.5299 = 180.166 \) meters.

Step2: Find segment from 193m side

We have another right triangle with hypotenuse 193 m and angle \( 21^\circ \). Let the length of the tunnel segment from the second entrance be \( y \). Using the cosine function:
\( \cos(21^\circ)=\frac{y}{193} \)
\( y = 193\times\cos(21^\circ) \)
Calculate \( \cos(21^\circ)\approx0.9336 \), so \( y\approx193\times0.9336 = 179.1848 \) meters.

Step3: Sum the two segments

The total length of the tunnel is \( x + y \).
\( x + y\approx180.166 + 179.1848 = 359.3508 \) meters.

Step4: Round to nearest tenth

Rounding \( 359.3508 \) to the nearest tenth gives \( 359.4 \) meters.

Answer:

\( 359.4 \) meters