Question

a woman hikes 505 m east, then turns 54° north of west and hikes 365 m. how far is the woman from her start?
? m
365 m
505 m 54°
round to the nearest hundredth.

Explanation:

Step1: Find internal triangle angle

The internal angle between the two sides is $180^\circ - 54^\circ = 126^\circ$.

Step2: Apply the Law of Cosines

Let $a=505$, $b=365$, $\theta=126^\circ$. The distance $c$ is:

$$ c = \sqrt{a^2 + b^2 - 2ab\cos(\theta)} $$

$$ c = \sqrt{505^2 + 365^2 - 2(505)(365)\cos(126^\circ)} $$

$$ c = \sqrt{255025 + 133225 - 368650\times(-0.5878)} $$

$$ c = \sqrt{388250 + 216692.47} = \sqrt{604942.47} $$

Answer:

$777.78$ m