(a) A sector of a circle of radius 8cm subtends an angle of 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 (pi = frac{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 ;
Explanation
(a) (i) Length of sector = (frac{theta}{360} times 2pi R)
Length of the base of the cone formed = (2pi r)
(frac{90}{360} times 2 times pi times 8 = 2pi r)
(4pi = 2pi r)
(2r = 4 implies r = 2 cm)
The base radius of the cone = 2 cm.
(ii) Height of the cone
(h^{2} = 8^{2} – 2^{2})
(h^{2} = 64 – 4 = 60)
(h = sqrt{60})
= (7.746 cm)
(iii) volume of a cone = (frac{1}{3} pi r^{2} l)
= (frac{1}{3} times frac{22}{7} times 2 times 2 times 7.746)
= (frac{88 times 7.746}{21})
= (32.46 cm^{3})
(b)(i) (100,000 cm = 1 km)
1 cm represents 20,000 metres
(therefore 4.5 cm = (4.5 times 20,000) cm)
= 90,000 cm
Converting to km,
= (frac{90,000}{100,000})
= 0.9km
(ii) 1 metre represents 20,000 metres
16km = 16000m
Distance on the map in metres = (frac{16000}{20000})
= 0.8m
(iii) Area of the forest on the map
= (frac{85 times (10^{5})^{2} cm^{2}}{(20000)^{2}})
= (frac{85 times 10^{10}}{4 times 10^{8}})
= (2125 cm^{2})