Question

A surveyor wants to know the length of a tunnel built through a mountain. According to her equipment, she is located 212 meters from one entrance of the

tunnel, at an angle of 57° to the perpendicular. Also according to her equipment, she is 119 meters from the other entrance of the tunnel, at an angle of 14° to
the perpendicular, Based on these measurements, fiad 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.

Answer 1

Answer: Approximately 2 * AC = 737.2 meters. Rounded to the nearest tenth, the answer is 737.2 meters.

Step-by-step explanation: To solve the problem, we can draw a diagram:

A

/|

/ |

/ | x

/ |

/ |

/θ |

/ |

/ |

/ |

/ |

/ |

B-----------C

y

where A and B are the entrances of the tunnel, C is the point in the middle of the tunnel that we want to find, and θ, x, and y are the angles and distances given in the problem. We want to find the length of AC.

Using trigonometry, we can find the distances BC and AB: tan(57°) = x / y => x = y * tan(57°)

tan(14°) = x / (y + 212) => x = (y + 212) * tan(14°)

Setting these two expressions for x equal, we get: y * tan(57°) = (y + 212) * tan(14°)

Solving for y, we get: y = 212 / (tan(57°) / tan(14°) - 1) ≈ 286.5 meters

Now we can use the law of sines to find the length of AC: sin(θ) / AC = sin(14°) / BC = sin(57°) / AB

Solving for AC, we get: AC = AB * sin(θ) / sin(57°) ≈ 368.6 meters

Therefore, the length of the entire tunnel is approximately 2 * AC = 737.2 meters. Rounded to the nearest tenth, the answer is 737.2 meters.