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.]

Answers (1)
G Gautam harsolia

From the above given figure

Circumference of 1st semicircle l_1 = \pi r_1 = 0.5\pi


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