(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 surf...
Question 1 Report
(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 ;
Answer Details
(a) When the sector is folded into a cone, the arc length of the sector becomes the circumference of the base, and the sector radius becomes the slant height \(l = 8\text{ cm}\).
Arc length \(= \dfrac{90}{360}\times 2\pi(8) = \tfrac{1}{4}\times 16\pi = 4\pi\text{ cm}\).
(i) Base circumference \(2\pi r = 4\pi \ \Rightarrow\ \mathbf{r = 2\text{ cm}}\).
(ii) Height \(h = \sqrt{l^2 - r^2} = \sqrt{8^2 - 2^2} = \sqrt{60} = \mathbf{7.75\text{ cm}}\) (2 d.p.).
(iii) Volume \(= \tfrac{1}{3}\pi r^2 h = \tfrac{1}{3}\times\tfrac{22}{7}\times 4 \times 7.746 = \mathbf{32.5\text{ cm}^3}\) (3 s.f.).
(b) Scale \(1 : 20{,}000\) means \(1\text{ cm}\) on the map \(= 20{,}000\text{ cm}\) on the ground.
(i) \(4.5\text{ cm}\) represents \(4.5\times 20{,}000 = 90{,}000\text{ cm} = 900\text{ m} = \mathbf{0.9\text{ km}}\).
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