Irveyor wants to know the length of a tunnel built through a mountain. According to his equipment, he is located 212 meters from one entrance of the
nel, at an angle of to the perpendicular. Also according to his equipment, he is 119 meters from the other entrance of the tunnel, at an angle of to
perpendicular. Based on these measurements, find the length of the entire tunnel.
not round any intermediate computations. Round your answer to the nearest tenth.
te that the figure below is not drawn to scale.

Ask by Erickson Ortega. in the United States

Mar 29,2025

Question

Irveyor wants to know the length of a tunnel built through a mountain. According to his equipment, he is located 212 meters from one entrance of the
nel, at an angle of to the perpendicular. Also according to his equipment, he is 119 meters from the other entrance of the tunnel, at an angle of to
perpendicular. Based on these measurements, find the length of the entire tunnel.
not round any intermediate computations. Round your answer to the nearest tenth.
te that the figure below is not drawn to scale.

Ask by Erickson Ortega. in the United States

Mar 29,2025

UpStudy AI · Accepted Answer

We begin by imagining the tunnel as a straight line (the horizontal direction) and the perpendicular as the vertical direction. Irveyor’s two measurements give distances from his position to the tunnel entrances. In each case the measured line makes a given angle with the perpendicular. In such a situation the horizontal (tunnel-axis) distance from Irveyor to an entrance is computed by taking the sine of the given angle times the distance. This is because in the right‐triangle formed by his line of sight, the component along the tunnel is opposite the given angle.

Let  
$$
d_1 = 212\sin(57^\circ)
$$
be the horizontal distance from his position to the first entrance, and  
$$
d_2 = 119\sin(14^\circ)
$$
be the horizontal distance to the other entrance. Since the tunnel is a straight line, the full length of the tunnel is the sum of these two distances:
$$
L = d_1 + d_2 = 212\sin(57^\circ) + 119\sin(14^\circ).
$$

Now, we compute each term without rounding intermediate results:

1. Compute $$\sin(57^\circ)$$.  
   (Using a calculator, $$\sin(57^\circ)$$ can be computed to high accuracy.)  
   $$
   \sin(57^\circ) \approx 0.8386705679.
   $$
   Then,  
   $$
   212\sin(57^\circ) \approx 212 \times 0.8386705679 \approx 177.773.
   $$

2. Compute $$\sin(14^\circ)$$.  
   $$
   \sin(14^\circ) \approx 0.2419218950.
   $$
   Then,  
   $$
   119\sin(14^\circ) \approx 119 \times 0.2419218950 \approx 28.787.
   $$

Adding the two distances gives:
$$
L \approx 177.773 + 28.787 \approx 206.560 \text{ meters}.
$$

Rounded to the nearest tenth, the length of the tunnel is:
$$
L \approx 206.6 \text{ meters}.
$$
The length of the entire tunnel is approximately 206.6 meters.

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