Class 10

Math

All topics

Arithmetic Progressions

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

Circumference of first semicircle $=$ $πr=0.5π$

Circumference of second semicircle $=$ $πr=π$

Circumference of third semicircle $=$ $πr=1.5π$

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

Hence, length of spiral can be calculated as follows:

$S=2n [2a+(n−1)d]$

$=$$213 (2×0.5π+12×0.5π)$

$=$ $213 ×7π$

$=$ $213 ×7×722 $

$=143$ cm