(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, \(\frac{3}{\sqrt{3}}(\frac{2}{\sqrt{3}} - \frac{\sqrt{12}}{6})\).

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(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

π = \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 c m = 21 c m

The chord AB =

2 r \sin \frac{θ}{2}

14 = 2 (21) \sin \frac{θ}{2}

\sin \frac{θ}{2} = \frac{14}{42} = 0.333

\frac{θ}{2} = \sin^{- 1} (0.333)

\frac{θ}{2} = 19.469 °

θ = 19.469 ° \times 2 = 38.938 °

Length of the arc =

\frac{θ}{360 °} \times 2 π r

\frac{38.938}{360} \times 2 \times \frac{22}{7} \times 21

14.277 c m

Perimeter of the remaining portion = 22 + 12 + 12 + 5 + 3 + 14.277
= 68.277 cm

≊

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.

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(a) A segment of a circle is cut off from a rectangular board as shown in the diagram. ...

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