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 = 43.9°. find the distance across the lake from a to b.

image of a triangle with a lake (blue shape) between points a, b, and c. side ac is 307 m, side bc is 453 m, angle at a (∠cab) is 43.9°

note: the triangle is not drawn to scale.

distance = m

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 \(a = 453\), \(b=307\), and \(C = 43.9^{\circ}\). We want to find the length of \(AB\) (let \(AB = c\)).
So, \(c^{2}=453^{2}+307^{2}-2\times453\times307\times\cos(43.9^{\circ})\).
First, calculate \(453^{2}=453\times453 = 205209\), \(307^{2}=307\times307=94249\).
Then, \(2\times453\times307 = 2\times139071=278142\).
\(\cos(43.9^{\circ})\approx0.720\).

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

\(c^{2}=205209 + 94249-278142\times0.720\).
\(c^{2}=205209+94249 - 200262.24\).
\(c^{2}=299458-200262.24\).
\(c^{2}=99195.76\).

Step3: Find \(c\)

Take the square - root of \(c^{2}\), \(c=\sqrt{99195.76}\approx314.95\).

Answer:

\(314.95\)