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 _ca_ = 47.3°. find the distance across the lake from a to b.

note: the triangle is not drawn to scale.
distance ≈ 665.44 × m
enter your answer as a number; your answer should be accurate to 2 decimal places.
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Explanation:

Step1: Apply the Law of Cosines

The Law of Cosines formula is \(c^{2}=a^{2}+b^{2}-2ab\cos C\). In \(\triangle ABC\), let \(a = 514\), \(b=291\), and \(C = 47.3^{\circ}\). We want to find the length of \(AB\) (let's call it \(c\)).

$$c^{2}=514^{2}+291^{2}-2\times514\times291\times\cos(47.3^{\circ})$$

Step2: Calculate each term

First, \(514^{2}=264196\), \(291^{2}=84681\).
\(\cos(47.3^{\circ})\approx0.678\)
\(2\times514\times291 = 2\times149574=299148\)
\(299148\times0.678\approx202822.34\)
Then \(514^{2}+291^{2}-2\times514\times291\times\cos(47.3^{\circ})=264196 + 84681-202822.34\)

$$264196+84681-202822.34=348877 - 202822.34=146054.66$$

Step3: Find \(c\)

Since \(c^{2}=146054.66\), then \(c=\sqrt{146054.66}\approx382.17\)

Answer:

\(382.17\)