Theory question 106
Mensuration of Plane Shapes(a) The scale of a map is 1: 20,000. Calculate the area, in square centimetres, on the map of a forest reserve which covers 85 km2.
(b) A rectangular playing field is 18m wide. It is surrounded by a path 6m wide such that its area is equal to the area of the path. Calculate the length of the field.
(c) The diagram shows a circle centre O. If ∠POQ = x°, the diameter of the circle is 7cm and the area of the shaded portion is 27.5 cm2, find, correct to the nearest degree, the value of x. [Take π = 22/7]
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(a) 1 cm represents 200 m, so 1 cm2 represents 0.04 km2. 85 ÷ 0.04 = 2125 cm2.
(b) Outer rectangle = (l + 12) × 30. Path area = 30(l + 12) − 18l = 12l + 360 = field area 18l, so 6l = 360 and l = 60 m.
(c) Circle area = 22/7 × 3.52 = 38.5 cm2. The shaded major sector has angle 360° − x: (360 − x)/360 × 38.5 = 27.5, so 360 − x ≈ 257.1 and x ≈ 103°.
Area scale is the square of the length scale.