VerifiedHint: Here we will proceed by assuming the lengths of semicircles as ${l_1},{l_2},{l_3},{l_4},......,{l_{13}}$ and then apply the formula of perimeter of semicircle i.e. $\pi r$ to calculate the length of each semicircle to get the total length of the spiral.
Complete step by step answer:
Let ${l_1},{l_2},{l_3},{l_4},......,{l_{13}}$ be the lengths of semicircles of radii 0.5cm, 1cm, 2cm …… and $\dfrac{{13}}{2}$ cm respectively.
Now we will apply the perimeter of semicircle= $\pi r$ to calculate the length of each semicircle.
${l_1} = \pi \times 0.5 = \dfrac{\pi }{2}cm$
${l_2} = \pi \times 1 = 2\left( {\dfrac{\pi }{2}} \right)cm$
${l_3} = \pi \times 1.5 = 3\left( {\dfrac{\pi }{2}} \right)cm$
${l_4} = \pi \times 2 = 4\left( {\dfrac{\pi }{2}} \right)cm$
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${l_{13}} = \pi \times \dfrac{{13}}{2} = 13\left( {\dfrac{\pi }{2}} \right)cm$
Therefore,
We will add all the lengths of the spiral,
Total length of the spiral= ${l_1} + {l_2} + {l_3} + {l_4} + ..... + {l_{13}}$
$ = \left[ {\dfrac{\pi }{2} + 2\dfrac{\pi }{2} + 3\dfrac{\pi }{2} + 4\dfrac{\pi }{2} + ..... + 13\dfrac{\pi }{2}} \right]cm$
$ = \dfrac{\pi }{2}\left( {1 + 2 + 3 + 4 + .... + 13} \right)cm$
$ = \dfrac{\pi }{2} \times \dfrac{{13}}{2} \times 14cm$
$ = \left( {\dfrac{1}{2} \times \dfrac{{22}}{7} \times \dfrac{{13}}{2} \times 14} \right)cm$
= 143cm
Hence the required length of the spiral is 143cm.
Note: We can solve this type of question by another way i.e. converting the lengths of semicircles into sequence of arithmetic progression and then use sum formula i.e. ${S_n} = \dfrac{n}{2}\left( {2a + \left( {n - 1} \right)d} \right)$ to get the total length of spiral.
