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A spiral is made up of successive semicircles, with centres alternately at A and B, starting with centre at A, of 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 semicircles? (Take $\pi = \frac{22}{7}$)
[Hint: Length of successive semicircles is $l_1
Given:
A spiral is made up of successive semicircles, with centres alternately at A and B, starting with centre at A, of radii 0.5 cm, 1.0 cm, 1.5 cm, 2.0 cm,…
To do:
We have to find the total length of such a spiral made up of thirteen consecutive semicircles.
Solution:
Here,
$R_{1}=0.5\ cm, R_{2}=1.0\ cm, R_{3}=1.5\ cm$
$a=0.5\ cm, d=1.0\ cm-0.5\ cm=0.5\ cm$
Length of the spiral $=$ Sum of perimeter of $13$ consecutive semicircles
$=\pi R_{1}+\pi R_{2}+\pi R_{3}+\ldots . .+\pi R_{13}$
$=\pi[R_{1}+R_{2}+R_{3}+\ldots . .R_{13}]$
$=\pi[0.5+1.0+1.5+\ldots .+6.5]$
$=\pi[\frac{13}{2}[2 \times 0.5+(13-1)(0.5)]]$
$=\pi[\frac{13}{2}(1+12 \times 0.5)]$
$=\pi \times(\frac{13}{2} \times 7)$
$=\frac{22}{7} \times \frac{13}{2} \times 7$
$=143\ cm$
The total length of the spiral made up of thirteen consecutive semicircles is 143 cm.