Step 1 :We are given a triangle ABC with AC = 3800m, BC = 3200m, and ∠C = 57°. We are asked to find the length of AB, which is the proposed tunnel.
Step 2 :We can use the Law of Cosines to solve this problem. The Law of Cosines states that for any triangle with sides of lengths a, b, and c and an angle γ opposite side c, the following equation holds: \(c² = a² + b² - 2ab\cos(γ)\).
Step 3 :In this case, a = AC = 3800m, b = BC = 3200m, and γ = ∠C = 57°. We can substitute these values into the Law of Cosines to find the length of AB.
Step 4 :Substituting the given values into the Law of Cosines, we get \(AB² = 3800² + 3200² - 2*3800*3200*\cos(57°)\).
Step 5 :Solving the above equation, we find that AB = 3381.5m.
Step 6 :Final Answer: The length of the tunnel is \(\boxed{3381.5}\) meters.