To measure the distance through a mountain for a proposed tunnel, a point $\mathrm{C}$ is chosen that can be reached from each end of the tunnel. See the figure to the right. If $A C=3800 \mathrm{~m}, B C=3200 \mathrm{~m}$, and angle $C=57^{\circ}$, find the length of the tunnel.
The length of the tunnel is $\mathrm{m}$.
(Do not round until the final answer. Then round to the nearest tenth as needed.)
Answer
Final Answer: The length of the tunnel is \(\boxed{3381.5}\) meters.
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.
