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 ∠cab = 52.2°. find the distance across the lake from a to b. note: the triangle is not drawn to scale. distance = ft enter your answer as a number; your answer should be accurate to 2 decimal places. question help: worked example 1

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 \(a = 543\), \(b=350\), and \(C = 52.2^{\circ}\). We want to find the length of \(AB\) (let's call it \(c\)).
So, \(c^{2}=543^{2}+350^{2}-2\times543\times350\times\cos(52.2^{\circ})\).
First, calculate \(543^{2}=543\times543 = 294849\), \(350^{2}=350\times350=122500\), and \(\cos(52.2^{\circ})\approx0.6131\).
Then \(2\times543\times350\times0.6131=(2\times543\times350)\times0.6131 = 380100\times0.6131\approx233031.31\).

Step2: Calculate \(c^{2}\)

\(c^{2}=294849 + 122500-233031.31\).
\(c^{2}=417349-233031.31=184317.69\).

Step3: Find \(c\)

Take the square - root of \(c^{2}\). \(c=\sqrt{184317.69}\approx429.32\).

Answer:

\(429.32\)