(a) The angles of depression of the top and bottom of a building are 51° and 62° respectively from the top of a tower 72m high. The base of the building is on the same horizontal level as the foot of the tower. Calculate the height of the building correct to 2 significant figures. (b)In the diagram, PR is a chord of the circle centre O and radius 30cm,< POR = 120°. Calculate correct to three significant figures : (i) the length of chord PR ; (ii) the length of arc PQR ; (iii) the perimeter of the shaded portion. (Take \(\pi = 3.142\)).

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(a) The angles of depression of the top and bottom of a building are 51° and 62° respectively from the top of a tower 72m high. The base of the building is on the same horizontal level as the foot of the tower. Calculate the height of the building correct to 2 significant figures.
(b) In the diagram, PR is a chord of the circle centre O and radius 30cm, < POR = 120°. Calculate correct to three significant figures : (i) the length of chord PR ; (ii) the length of arc PQR ; (iii) the perimeter of the shaded portion. (Take

π = 3.142

Explanation

(a)
ST = QR ; PS = height of tower.
In

Δ

PST,

\frac{72}{S T} = \tan 62 °

S T = \frac{72}{\tan 62}

= 38.283m

≊

38.3m
In

Δ

PQR,

\frac{h}{Q R} = \tan 51 °

h = Q R \times \tan 51 °

38.3 \times 1.235

47.3 m

QS = RT (height of the building)
QS = PS - PQ
= 72 - 47.3 = 24.7m
(b)
(i)

Δ P O R = Δ Q O R

(isosceles triangle)

\frac{P T}{30} = \sin 60

P T = 30 \sin 60 = 30 \times \frac{\sqrt{3}}{2}

15 \sqrt{3} c m

PT = TR
PR = 2PT
=

2 \times 15 \sqrt{3} = 30 \sqrt{3} c m

30 \times 1.732 = 51.96 c m

≊ 52 c m

(ii) Length of minor arc, PQR =

\frac{θ}{360} \times 2 π r

\frac{120}{360} \times 2 \times 3.142 \times 30

62.84 c m

(iii) Perimeter of shaded portion = chord PR + Arc PQR
= 51.96 + 62.84
= 114.8 cm

≊

115cm.

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(a) The angles of depression of the top and bottom of a building are 51° and ...

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