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A spiral is made up of successive semicircles, with centers alternately at $$A$$ and $$B$$, starting with center at $$A$$, of radii $$0.5\ cm, 1.0\ cm,1.5\ cm,2.0\ cm$$, . . .  as shown in Fig. What is the total length of such a spiral made up of thirteen consecutive semicircles
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Solution

Circumference of first semicircle $$=$$ $$\pi r=0.5\pi $$

Circumference of second semicircle $$=$$ $$\pi r=\pi $$

Circumference of third semicircle $$=$$ $$\pi r=1.5\pi $$

It is clear that $$a=0.5\pi $$, $$d=0.5\pi $$ and $$n = 13$$

Hence, length of spiral can be calculated as follows:

$$S=\dfrac{n}{2}\left [ 2a+(n-1)d \right ]$$

$$=$$$$\dfrac{13}{2}(2\times 0.5\pi +12\times 0.5\pi )$$

$$=$$ $$\dfrac{13}{2}\times 7\pi$$

$$=$$ $$\dfrac{13}{2}\times 7\times \dfrac{22}{7}$$

$$=143$$ cm

Mathematics
RS Agarwal
Standard X

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