a. In a town, Chief X resides 60 m away on a bearing of 057° from Palace P, while Chief Y resides on a bearing of 150° from the same Palace P. The residence of X and Y are 180 m apart. Illustrate the information in a diagram.
b. Find and correct to three significant figures, the: i. bearing of X from Y; ii. distance between P and Y.
Explanation
a. the diagram above solution to question a.
bi. Using the sine rule
\(\frac{sinP}{XY} = \frac{sinY}{PX}\)
= \(\frac{sin93°}{180} = \frac{sinY}{60}\)
cross multiply
\(sinY\times180 = sin93°\times60\)
sinY = 0.3329
Y = \(sin^{-1}0.3329\)
Y = 19.44°
the angle between 30° and 19.44° at Y = 30° - 19.44° = 10.56°
Therefore, the bearing X from Y = 360° - 10.56° = 349°( to 3 sig. figure)
bii. Using cosine rule
Let a = |PY|, b = |PX| and c = |XY|
\(a^2 = b^2 + c^2 - 2bcCos A\) where A = 67.56°
\(a^2 = 60^2 + 180^2 - 2(60)(180)cos 67.56^0\)
\(a2 = 3600 + 32400 - 8245.06\)
\(a^2 = 27754.94^0\)
a = \(\sqrt{27754.94}\)
a = 166.6 m
∴ |PY| = 167 m (to 3 s.f)