(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)).
Explanation
(a)
ST = QR ; PS = height of tower.
In (Delta) PST,
(frac{72}{ST} = tan 62°)
(ST = frac{72}{tan 62})
= 38.283m (approxeq) 38.3m
In (Delta) PQR, (frac{h}{QR} = tan 51°)
(h = QR times tan 51°)
= (38.3 times 1.235)
= (47.3m)
QS = RT (height of the building)
QS = PS – PQ
= 72 – 47.3 = 24.7m
(b)
(i) (Delta POR = Delta QOR) (isosceles triangle)
(frac{PT}{30} = sin 60)
(PT = 30 sin 60 = 30 times frac{sqrt{3}}{2})
= (15sqrt{3} cm)
PT = TR
PR = 2PT
= (2 times 15sqrt{3} = 30sqrt{3} cm)
= (30 times 1.732 = 51.96 cm)
(approxeq 52 cm)
(ii) Length of minor arc, PQR = (frac{theta}{360} times 2pi r)
= (frac{120}{360} times 2 times 3.142 times 30)
= (62.84 cm)
(iii) Perimeter of shaded portion = chord PR + Arc PQR
= 51.96 + 62.84
= 114.8 cm (approxeq) 115cm.