(a)
A segment of a circle is cut off from a rectangular board as shown in the diagram. If the radius of the circle is (1frac{1}{2}) times the length of the chord; calculate, correct to 2 decimal places, the perimeter of the remaining portion. [Take (pi = frac{22}{7})]
(b) Evaluate without using calculators or tables, (frac{3}{sqrt{3}}(frac{2}{sqrt{3}} – frac{sqrt{12}}{6})).
Explanation
(a) Length of the chord = 14 cm
Radius = (frac{3}{2} times 14 cm = 21 cm)
The chord AB = (2r sin frac{theta}{2})
(14 = 2(21) sin frac{theta}{2})
(sin frac{theta}{2} = frac{14}{42} = 0.333)
(frac{theta}{2} = sin^{-1} (0.333))
(frac{theta}{2} = 19.469°)
(theta = 19.469° times 2 = 38.938°)
Length of the arc = (frac{theta}{360°} times 2pi r)
(frac{38.938}{360} times 2 times frac{22}{7} times 21)
= (14.277 cm)
Perimeter of the remaining portion = 22 + 12 + 12 + 5 + 3 + 14.277
= 68.277 cm
(approxeq) 68.28 cm (2 decimal places).
(b) (frac{3}{sqrt{3}}(frac{2}{sqrt{3}} – frac{sqrt{12}}{6})
= (frac{3}{sqrt{3}}(frac{12 – sqrt{36}}{6sqrt{3}})
= (frac{3}{sqrt{3}}(frac{12 – 6}{6sqrt{3}})
= (frac{3}{sqrt{3}}(frac{6}{6sqrt{3}})
= (frac{18}{6sqrt{9}})
= (frac{18}{18})
= 1.