Question

points a and b are separated by a lake. to find the distance between them, a surveyor locates a point c on land such than \\( \angle c a b=48.7^{circ} \\). find the distance across the lake from a to b. note: the triangle is not drawn to scale. distance \\( \approx \\) m enter your answer as a number; your answer should be accurate to 2 decimal places. question help: \\( \square \\) post to forum

Explanation:

Step1: Apply the Law of Cosines

The Law of Cosines states that for a triangle with sides \(a\), \(b\), \(c\) and the angle \(C\) opposite side \(c\), \(c^{2}=a^{2}+b^{2}-2ab\cos C\). In \(\triangle ABC\), let \(a = 529\), \(b=380\), and \(C = 48.7^{\circ}\). We want to find the length of side \(AB\) (let's call it \(c\)).

$$c^{2}=529^{2}+380^{2}-2\times529\times380\times\cos(48.7^{\circ})$$

Step2: Calculate each term

First, calculate \(529^{2}=529\times529 = 279841\), \(380^{2}=380\times380=144400\).
\(\cos(48.7^{\circ})\approx0.6604\)
\(2\times529\times380\times\cos(48.7^{\circ})=2\times529\times380\times0.6604=(2\times529\times380)\times0.6604 = 402040\times0.6604\approx265427.42\)

Step3: Find \(c^{2}\)

\(c^{2}=279841 + 144400-265427.42=358813.58\)

Step4: Find \(c\)

Take the square - root of \(c^{2}\), \(c=\sqrt{358813.58}\approx599.01\)

Answer:

\(599.01\)