(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 \(1\frac{1}{2}\) 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 = \(\frac{\theta}{360°} \times 2\pi r\)
\(\frac{38.938}{360} \times 2 \times \frac{22}{7} \times 21\)
= \(14.277 cm\)
Perimeter of the remaining portion = 22 + 12 + 12 + 5 + 3 + 14.277
= 68.277 cm
\(\approxeq\) 68.28 cm (2 decimal places).
(b) \(\frac{3}{\sqrt{3}}(\frac{2}{\sqrt{3}} - \frac{\sqrt{12}}{6}\)
= \(\frac{3}{\sqrt{3}}(\frac{12 - \sqrt{36}}{6\sqrt{3}}\)
= \(\frac{3}{\sqrt{3}}(\frac{12 - 6}{6\sqrt{3}}\)
= \(\frac{3}{\sqrt{3}}(\frac{6}{6\sqrt{3}}\)
= \(\frac{18}{6\sqrt{9}}\)
= \(\frac{18}{18}\)
= 1.