Ex 5.3, 18 - A spiral is made up of successive semicircles - Ex 5.3

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Ex 5.3 ,18 A spiral is made up of successive semicircles, with centres alternately at A and B, starting with centre at A of radii 0.5, 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 = 22/7) Here we have to find total length, i.e., total circumference of all semicircles We know that Circumference of circle = 2 r So, circumference of semicircle = 1/2 2 r = r Circumference of 1st semi circle = 0.5 Circumference of 2nd semicircle = 1.0 Circumference of 3rd semicircle = 1.5 . Circumference of 13th semicircle = 6.5 So, the series is 0.5, 1.0, 1.5, ., 6.5 Taking common = (0.5, 1.0, 1. 5, 2 .6.5) = S Where S is sum of series 0.5, 1.0, 1.5, 2 .6.5 For 0.5, 1.0, 1. 5, 2 .6.5 As difference is same, it is an AP S can be found by using formula S = /2 ( + ) Here n = 13, a = 0.5, = 6.5 Putting values in formula S = 13/2 (0.5+6.5) S = 13/2 7 S = 91/2 S = 45.5 Length of spiral = Sum of series = S = 45.5 = 22/7 45.5 = 22 6.5 = 143 Length of spiral is 143 cm

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