Theory question 103
Mensuration of Plane Shapes(a) The angle of a sector of a circle radius 7cm is 108°. Calculate the perimeter of the sector (Take π = 22/7).
(b) A boat is on the same horizontal level as the foot of a cliff, and the angle of depression of the boat from the top of the cliff is 30°. If the boat is 120m away from the foot of the cliff, find the height of the cliff correct to three significant figures.
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Model answer
(a) Arc length = (108/360) × 2 × 22/7 × 7 = 0.3 × 44 = 13.2 cm
Perimeter = arc + two radii = 13.2 + 7 + 7 = 27.2 cm
(b) Angle of elevation of the top from the boat = angle of depression = 30°
tan 30° = h/120, so h = 120 × tan 30° = 120 × 0.5774 = 69.28
Height ≈ 69.3 m (3 s.f.)
Explanation
The perimeter of a sector includes the two radii as well as the arc.