(a) Without using tables, find the value of \(\frac{0.45 \times 0.91}{0.0117}\) (b) Find the number which is exactly halfway between \(1\frac{6}{7}\) and \(2\frac{11}{28}\). (c) If each interior angle of a regular polygon is five times the exterior angle, how many sides has the polygon? (d) Calculate the volume of the material used in making a pipe 20cm long, with an internal diameter 6cm and external diameter 8cm. [Take \(pi = \frac{22}{7}\)].
Explanation
(a) \(\frac{0.45 \times 0.91}{0.0117} = \frac{45 \times 91}{117}\) = \(5 \times 7 = 35\) (b) Halfway between \(1\frac{6}{7}\) and \(2\frac{11}{28}\) = \(\frac{1\frac{6}{7} + 2\frac{11}{28}}{2}\) = \((\frac{13}{7} + \frac{67}{28}) \times \frac{1}{2}\) = \(\frac{119}{28} \times \frac{1}{2} \) = \(\frac{119}{56} = 2\frac{7}{56} = 2\frac{1}{8}\) (c) Let the interior angle = 5x° and the exterior angle = x°. \(\therefore 5x° + x° = 180°\) \(6x° = 180° \implies x = 30°\) Exterior angle = \(\frac{360°}{n} \implies 30° = \frac{360°}{n}\) \(n = \frac{360°}{30°} = 12\) The polygon has 12 sides. (d) Length of pipe = 20 cm Internal diameter = 6 cm ; radius = 3cm External diameter = 8 cm ; radius = 4cm Let the external radius = R and internal radius = r \(\therefore\) Volume of material used = \(\pi R^{2} l - \pi r^{2} l\) Where l = length. \(Volume = \pi l (R^{2} - r^{2}) cm^{3}\) = \(\frac{22}{7} \times 20 (4^{2} - 3^{2}) = \frac{22}{7} \times 20 \times 7 = 440 cm^{3}\)