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 tothree 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)

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