Question

# 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

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:$$S=\dfrac{n}{2}\left [ 2a+(n-1)d \right ]$$$$=$$$$\dfrac{13}{2}(2\times 0.5\pi +12\times 0.5\pi )$$$$=$$ $$\dfrac{13}{2}\times 7\pi$$$$=$$ $$\dfrac{13}{2}\times 7\times \dfrac{22}{7}$$$$=143$$ cmMathematicsRS AgarwalStandard X

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