Question

A spiral is made up of successive semicircles, with centres alternately at $A$ and $B$, starting with centre at $A$, of radii $0.5cm,1cm,1.5cm,2cm,.....$ as shown in figure. What is the total length of such a spiral made up of thirteen consecutive semicircles? $(Take\pi =\frac{22}{7})$

Answer 1

$\Rightarrow$Let $l_1,l_2,l_3,...........$ be the lengths of semicircles of radii.

$\Rightarrow$$r_1=0.5cm,r_2=1cm,r_3=1.5cm,$ ......... respectively.

$\Rightarrow$We know that, circumference of a circle $=2\pi r$

$\therefore$ Length of a semicircle $=\pi r$

$\Rightarrow$$l_1=\pi r_1=\pi \times 0.5=\frac{\pi }{2},l_2=\pi r_2=\pi \times 1=\pi =2\left(\frac{\pi }{2}\right)$ and so on

$\Rightarrow$Similarly, $l_{13}=\pi r_{13}=\pi \times \frac{13}{2}=13\left(\frac{\pi }{2}\right)$

$\Rightarrow$Total length of spiral $=l_1+l_2+..........+l_{13}$

$=\frac{\pi }{2}+2\left(\frac{\pi }{2}\right)+3\left(\frac{\pi }{2}\right)+........+13\left(\frac{\pi }{2}\right)=\frac{\pi }{2}[1+2+3+..........+13]$

$=\frac{\pi }{2}\left[\frac{13(13+1)}{2}\right]=\frac{\pi }{2}\times \frac{13\times 14}{2}=\frac{1}{2}\times \frac{22}{7}\times 13\times 7=143cm.$