rveyor wants to know the length of a tunnel built through a mountain. According to his equipment, he is located 325 meters from one entrance of the
nel, at an angle of to the perpendicular. Also according to his equipment, he is 230 meters from the other entrance of the tunnel, at an angle of
perpendicular. Based on these measurements, find the length of the entire tunnel.
not round any intermediate computations. Round your answer to the nearest tenth.
the that the figure below is not drawn to scale.

Ask by Goodwin Reid. in the United States

Mar 29,2025

Question

rveyor wants to know the length of a tunnel built through a mountain. According to his equipment, he is located 325 meters from one entrance of the
nel, at an angle of to the perpendicular. Also according to his equipment, he is 230 meters from the other entrance of the tunnel, at an angle of
perpendicular. Based on these measurements, find the length of the entire tunnel.
not round any intermediate computations. Round your answer to the nearest tenth.
the that the figure below is not drawn to scale.

Ask by Goodwin Reid. in the United States

Mar 29,2025

UpStudy AI · Accepted Answer

Let the tunnel be a straight horizontal line. The surveyor’s position is off the tunnel at a fixed perpendicular distance, say $$ d_p $$. From his position, drop a perpendicular to the tunnel meeting it at a point $$ M $$. Let the distances along the tunnel from $$ M $$ to the two entrances be $$ x $$ and $$ L-x $$, where $$ L $$ is the length of the tunnel.

For the first entrance, the surveyor’s equipment gives a line‐of‐sight distance
$$
d_1 = 325\text{ m}
$$
making an angle of $$ 48^\circ $$ to the perpendicular. In the right triangle formed the side adjacent to the $$ 48^\circ $$ angle is the perpendicular distance $$ d_p $$ and the side opposite the angle is $$ x $$. Thus,
$$
d_p = 325 \cos 48^\circ \quad\text{and}\quad x = 325 \sin 48^\circ.
$$

For the second entrance, the equipment gives a line‐of‐sight distance
$$
d_2 = 230\text{ m}
$$
making an angle of $$ 19^\circ $$ to the perpendicular. Similarly, in this right triangle the perpendicular distance is
$$
d_p = 230 \cos 19^\circ,
$$
and the distance along the tunnel from $$ M $$ to this entrance is
$$
L-x = 230 \sin 19^\circ.
$$

Since the perpendicular distance $$ d_p $$ must be the same from both measurements, we have
$$
325 \cos 48^\circ = 230 \cos 19^\circ,
$$
which (approximately) holds for the given numbers.

Thus the total tunnel length is the sum of the horizontal components:
$$
L = x + (L-x) = 325 \sin 48^\circ + 230 \sin 19^\circ.
$$

Now, compute the numerical values (without rounding intermediate steps):

1. Calculate $$ 325 \sin 48^\circ $$:
   $$
   325 \sin 48^\circ \approx 325 \times 0.743144825 \approx 241.5211\,\text{m}.
   $$
2. Calculate $$ 230 \sin 19^\circ $$:
   $$
   230 \sin 19^\circ \approx 230 \times 0.325568154 \approx 74.8769\,\text{m}.
   $$

Therefore,
$$
L \approx 241.5211 + 74.8769 \approx 316.3980\,\text{m}.
$$

Rounded to the nearest tenth, the length of the tunnel is approximately $$ \boxed{316.4\,\text{m}} $$.
The length of the tunnel is approximately 316.4 meters.

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