Question

a bridge across a small bay is 300 meters long. from the west end, a surveyor measures and finds that the angle between the bridge and the island at b is 35°. from the east end a surveyor measures and finds that the angle between the bridge and the island at b is 53° how far is it from the west end of the bridge to the island? how far is it from the west end of the bridge to the island? 215.5 m 172.2 m 239.7 m 417.7 m

Explanation:

Step1: Find the angle at B

In triangle \(ABC\), using the angle - sum property of a triangle (\(A + B + C=180^{\circ}\)). Given \(A = 35^{\circ}\) and \(C = 53^{\circ}\), then \(B=180^{\circ}-(35^{\circ}+53^{\circ})=92^{\circ}\).

Step2: Apply the Law of Sines

The Law of Sines states that \(\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}\). We want to find the length from the west - end (\(A\)) to the island (\(B\)), i.e., \(BC\). Let \(AC = 300\) (side \(b\)), \(A = 35^{\circ}\), \(B = 92^{\circ}\), \(C = 53^{\circ}\). Using \(\frac{BC}{\sin A}=\frac{AC}{\sin B}\).
Substitute the values: \(BC=\frac{AC\times\sin A}{\sin B}\).
\(\sin A=\sin35^{\circ}\approx0.5736\), \(\sin B=\sin92^{\circ}\approx0.9994\), \(AC = 300\).
\(BC=\frac{300\times0.5736}{0.9994}\approx172.2\)

Answer:

172.2 m