A researcher wants to determine the width of a pond from east to west, which cannot be done by actual measurement. From a point $P$, she finds the distance to the eastern-most point of the pond to be $8 \; km$, while the distance to the western most point from $P$ to be $6 \; km$. If the angle between the two lines of sight is $60^\circ$, find the width of the pond.
In the figure,
eastern-most point of the pond $= E$
western-most point of the pond $= W$
width of the pond $= WE = p$
point of observation $= P$
distance to the eastern-most point of the pond from P $= PE = w = 8 \; km$
distance to the western-most point of the pond from P $= PW = e = 6 \; km$
$\angle WPE = 60^\circ$
Applying cosine rule to $\triangle WPE$ we have,
$\cos P = \dfrac{e^2 + w^2 - p^2}{2 e w}$
i.e. $\;$ $\cos \left(60^\circ\right) = \dfrac{6^2 + 8^2 - p^2}{2 \times 6 \times 8}$
i.e. $p^2 = 36 + 64 - 96 \cos 60^\circ$
i.e. $p^2 = 100 - \left(96 \times \dfrac{1}{2}\right) = 52$
$\therefore \;$ $p = \sqrt{52} = 2 \sqrt{13}$
$\therefore \;$ Width of the pond $= 2 \sqrt{13} \; km$