Theory question 106
Mensuration of Solids(a) A sector of a circle of radius 8 cm subtends an angle 90° at the centre of the circle. If the sector is folded without overlap to form the curved surface of a cone, find the: (i) base radius; (ii) height; (iii) volume, of the cone. (Take π = 22/7)
(b) A map is drawn to a scale of 1:20,000. Use it to calculate the: (i) distance, in kilometres, represented by 4.5 cm on the map; (ii) distance, in metres, on the map, between two towns 16 km apart; (iii) area on the map, of a forest, which covers 85 km2.
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(a)(i) Arc length = 90/360 × 2π × 8 = 4π = circumference of base 2πr, so r = 2 cm.
(ii) Slant height 8 cm: h = √(82 − 22) = √60 ≈ 7.75 cm.
(iii) V = ⅓ × 22/7 × 22 × 7.746 ≈ 32.5 cm3.
(b)(i) 4.5 × 20,000 = 90,000 cm = 0.9 km.
(ii) 16 km = 1,600,000 cm; 1,600,000 ÷ 20,000 = 80 cm = 0.8 m.
(iii) 1 cm represents 200 m, so 1 cm2 represents 0.04 km2. 85 ÷ 0.04 = 2125 cm2.
The arc of the sector becomes the circumference of the cone's base; the sector radius becomes the slant height.