Question

A spiral is made up of successive semicircles, with centres alternately at A and B, starting with centre A, for radii $0.5cm,1cm,1.5cm,2cm$, ...... as shown in the figure. What is the total length of such spiral made up of thirteen consecutive semicircles?

Answer 1

Perimeter of first semi-circle $=\pi \times r=\frac{22}{7}\times \frac{1}{2}=\frac{11}{7}$

Perimeter of second semi-circle $=\frac{22}{7}\times 1=\frac{22}{7}$

So, $d=\frac{22}{7}-\frac{11}{7}=\frac{11}{7}$

$S_{13}=\frac{13}{2}(2\times \frac{11}{7}+\frac{11}{7}\times 12)$

$=\frac{13}{2}\left(\frac{22+132}{7}\right)=\frac{13}{2}\times \frac{154}{7}=143cm$