Question: Michael wants to know the length of a tunnel built through a mountain. To do so, he makes the…

Michael wants to know the length of a tunnel built through a mountain. To do so, he makes the measurements shown in the figure below. Use these measurements to find the length of the tunnel.

[Diagram of a mountain with a tunnel through it. Two points on the mountain are labeled with distances: 258 m and 123 m. An angle of 82 degrees is marked between the two distances.]

Carry your intermediate computations to at least four decimal places. Round your answer to the nearest tenth of a meter.

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Solution

To find the length of the tunnel using the Law of Cosines, follow these steps: Given: - Side \( a = 123 \, \text{m} \) - Side \( c = 258 \, \text{m} \) - Angle \( \theta = 57^\circ \) Step 1: Apply the Law of Cosines \[ b^2 = a^2 + c^2 - 2ac \cos \theta & \] Step 2: Substitute the known values \[ b^2 = 123^2 + 258^2 - 2 \times 123 \times 258 \times \cos 57^\circ & \] Step 3: Calculate the squares \[ b^2 = 15129 + 66564 - 2 \times 123 \times 258 \times \cos 57^\circ & \] Step 4: Compute the product \( 2 \times 123 \times 258 \) \[ 2 \times 123 \times 258 = 63468 & \] Step 5: Calculate \( \cos 57^\circ \) \[ \cos 57^\circ \approx 0.544639 & \] Step 6: Multiply the results \[ 63468 \times 0.544639 \approx 34576.8311 & \] Step 7: Subtract the product from the sum of the squares \[ b^2 = 15129 + 66564 - 34576.8311 = 47016.1689 & \] Step 8: Take the square root to find \( b \) \[ b = \sqrt{47016.1689} \approx 216.7 \, \text{m} & \] Final Answer: 216.7 meters