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Q : 18 A spiral is made up of successive semicircles, with centres alternately at $\small A$ and $\small B$ , starting with centre at $\small A$ , of radii $\small 0.5\hspace {1mm}cm,1.0\hspace {1mm}cm,1.5\hspace {1mm}cm,2.0\hspace {1mm}cm,...$ as shown in Fig. $\small 5.4$ . 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 $\small l_1,l_2,l_3,l_4,...$ with centres at   $\small A,B,A,B,...,$ respectively.]

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best_answer

From the above given figure

Circumference of 1st semicircle $l_1 = \pi r_1 = 0.5\pi$
Similarly,

Circumference of 2nd semicircle $l_2 = \pi r_2 = \pi$

Circumference of 3rd semicircle $l_3 = \pi r_3 = 1.5\pi$

It is clear that this is an AP with $a = 0.5\pi \ and \ d = 0.5\pi$
Now, sum of length of 13 such semicircles is given by

$S_{13} = \frac{13}{2}\left \{ 2\times 0.5\pi + (13-1)0.5\pi\right \}$
$S_{13} = \frac{13}{2}\left ( \pi+6\pi \right )$
$S_{13} = \frac{13}{2}\times 7\pi$
$S_{13} = \frac{91\pi}{2} = \frac{91}{2}\times \frac{22}{7}=143$
Therefore, sum of length of 13 such semicircles is 143 cm

Posted by

Gautam harsolia

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