Question

A ship starts at point P, travels 200 miles to point Q adjusts its route according to the angle shown, and continues another 227 miles to point R. To the nearest mile, what is the distance from the starting position of the ship to its current position at point R?

Answer 1

Cosine rule:

$\begin{gathered} \Delta ABC \\ \\ a^2=b^2+c^2-2bc\cdot\cos A \end{gathered}$

For the given triangle

$PR^2=PQ^2+QR^2-2(PQ)(QR)\cdot\cos Q$$\begin{gathered} PR^2=200^2+227^2-2(200)(227)\cdot\cos 119 \\ \\ PR=\sqrt[]{200^2+227^2-2(200)(227)\cdot\cos 119} \\ \\ PR=\sqrt[]{40000+51529-90800\cos 119} \\ \\ PR=\sqrt[]{91529-90800\cdot\cos 119} \\ \\ PR\approx368 \end{gathered}$Then, the distance from starting point to point R is 368miles