The feet of two vertical poles of heights 3metres and 7metres are in line with a point P on the ground, the smaller pole being between the taller pole and P at a distance 20metres from P. The angle of elevation of the top (T) of the taller pole from the top (R) of the smaller pole is 30◦. Calculate the: (i) distance RT (ii) distance of the foot of the taller pole from P, correct to 3s.fs. (iii) angle of elevation of T from P, correct to 1d.p. Please with full solution
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(i) Distance RT
Let RT = x
Using the cosine rule,
x² = 20² + 7² - 2(20)(7)cos30°
x² = 400 + 49 - 280cos30°
x² = 449 - 140
x = √309
RT = 17.6m
(ii) Distance of the foot of the taller pole from P
Let TP = y
Using the cosine rule,
y² = 20² + 3² - 2(20)(3)cos30°
y² = 400 + 9 - 120cos30°
y² = 409 - 72
y = √337
TP = 18.3m
(iii) Angle of elevation of T from P
Let ∠TPT = θ
Using the sine rule,
sinθ = (7/18.3)sin30°
sinθ = 0.38
θ = 22.3°