A spiral is made up of successive semicircles, with centers alternately at A and B, starting with center at A, of radii 0.5 cm, 1.0 cm, 1.5 cm, 2 cm,....... as shown in Fig. What is the total length of such a spiral made up of thirteen consecutive semicircles?

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132 cm

143 cm

151 cm

165 cm

answer is B.

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Detailed Solution

Perimeter of first semicircle =

π r = \frac{22}{7} \times 0.5 = \frac{11}{7} cm

Perimeter of second semicircle =

π r = \frac{22}{7} \times 1 = \frac{22}{7} cm

Thus, perimeter forms an AP with a =

\frac{11}{7}

and d =

\frac{11}{7}

. For n = 13.

S_{n} = \frac{n}{2} [2 a + (n - 1) d

\Rightarrow S_{13} = \frac{13}{2} [2 \times \frac{11}{7} + (13 - 1) \frac{11}{7}] = \frac{13}{2} \times 14 \times \frac{11}{7} = 143 cm

So, Option 2 is correct.

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