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At the beginning of a carrom game, the coins are arranged in the central circle of the board as shown. What is the area in square inches in nearest integer of the central circle not occupied by the coins if the radius of each coin is $\frac{1}{2}$ inch.

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5.71 sq. inches

2.71 sq. inches

3.71 sq. inches

4.71 sq. inches

answer is D.

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Detailed Solution

Given,
Radius of each coin

r_{1} = \frac{1}{2}

inch

In the figure it can be seen that central circle radius is 5 times the radius of each coin. Central circle radius,

r_{2} = 5 \times r_{1}

r_{2} = 5 \times \frac{1}{2}

r_{2} = \frac{5}{2}

inch
Overall coins in central circle =19
Thus, the area of 19 circle,
=

19 \times π r_{1}^{2}

= 19 \times π \times {(\frac{1}{2})}^{2}

= \frac{19}{4} π

sq. inches
Area of central circle,

= π \times (\frac{5^{2}}{2})

= \frac{25}{4} π

Now, the area of central circle which is not occupied by the coins,

A = \frac{25}{4} π - \frac{19}{4} π

= \frac{6 π}{4}

A = 4.71

sq. inches
Correct option is 4.

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