A spiral is made up of successive semicircles | vedclass

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(C) The length of a semicircle is given by $l = \pi r$.For the given radii $r_1 = 0.5 \, cm, r_2 = 1.0 \, cm, r_3 = 1.5 \, cm, \ldots$,the lengths of

Vedclass · Accepted Answer

$143$

Vedclass · Answer

(C) The length of a semicircle is given by $l = \pi r$.For the given radii $r_1 = 0.5 \, cm, r_2 = 1.0 \, cm, r_3 = 1.5 \, cm, \ldots$,the lengths of the successive semicircles are:$l_1 = \pi(0.5) = 0.5\pi \, cm$$l_2 = \pi(1.0) = 1.0\pi \, cm$$l_3 = \pi(1.5) = 1.5\pi \, cm$These lengths form an Arithmetic Progression $(A.P.)$ where the first term $a = 0.5\pi$ and the common difference $d = 1.0\pi - 0.5\pi = 0.5\pi$.We need to find the sum of $n = 13$ terms.The sum of $n$ terms of an $A.P.$ is given by $S_n = \frac{n}{2} [2a + (n - 1)d]$.Substituting the values:$S_{13} = \frac{13}{2} [2(0.5\pi) + (13 - 1)(0.5\pi)]$$S_{13} = \frac{13}{2} [1.0\pi + 12(0.5\pi)]$$S_{13} = \frac{13}{2} [1.0\pi + 6.0\pi] = \frac{13}{2} [7\pi]$$S_{13} = \frac{91\pi}{2}$Using $\pi = \frac{22}{7}$:$S_{13} = \frac{91}{2} 	imes \frac{22}{7} = 13 	imes 11 = 143 \, cm$.Thus,the total length of the spiral is $143 \, cm$.

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