Question

A spiral is made up of successive semi-circles, with centres alternately at A and B, starting with centre at radii $0.5$ cm, $1.0$ cm, $1.5$ cm, $2.0$ cm, as shown in figure, what is the total length of such a spiral made up of thirteen consecutive semi-circles?
(Take $\pi$ $=\frac{22}{7}$)

• A. $143$ cm
  
• B. $171$ cm
  
• C. $190$ cm
  
• D. $222$ cm
  

Answer 1

The correct option is A $143$ cm

Semi-perimeter of circle $=\pi r$
$l_1=\pi \left(0.5\right)=\frac{\pi }{2}cm.$
$l_2=\pi \left(1\right)=\pi$
$l_3=\pi \left(1.5\right)=\frac{3\pi }{2}$
$l_1,l_2,l_3$ the length of semicircle are in A.P
$\frac{\pi }{2},\pi ,\frac{3\pi }{2},2\pi ......$
Hence $a=\frac{\pi }{2},d=\frac{\pi }{2}$

$S_{13}=\frac{13}{2}\left[2\right(\frac{\pi }{2})+(13-1\left)\right(\frac{\pi }{2}]$

$\Rightarrow \frac{13}{2}[\pi +\frac{12\pi }{2}$

$\Rightarrow \frac{13}{2}\times 7\pi$

$\Rightarrow \frac{91\pi }{2}$

$\Rightarrow \frac{91\times 22}{7\times 2}$

$\Rightarrow 143$