Last updated at April 8, 2021 by Teachoo
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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, โฆโฆโฆ as shown in figure. What is the total length of such a spiral made up of thirteen consecutive semicircles? (take ๐ = 22/7) Here we have to find total length, i.e., total circumference of all semicircles We know that Circumference of circle = 2ฯr So, Circumference of semicircle = 1/2 ร 2ฯr = ฯr Circumference of 1st semi circle = ฯ ร 0.5 Circumference of 2nd semicircle = ฯ ร 1.0 Circumference of 3rd semicircle = ฯ ร 1.5 โฆ. Circumference of 13th semicircle = ฯ ร 6.5 So, the series is ฯ ร 0.5, ฯ ร 1.0, ฯ ร 1.5, โฆโฆโฆโฆโฆ., ฯ ร 6.5 Taking ๐ common = ๐ (0.5, 1.0, 1. 5, 2 โฆโฆ.6.5) = ๐ S Where S is sum of series 0.5, 1.0, 1.5, 2 โฆโฆ.6.5 For 0.5, 1.0, 1. 5, 2 โฆโฆ.6.5 As difference is same, it is an AP S can be found by using formula S = ๐/๐ (๐+๐) Putting n = 13, a = 0.5, ๐ = 6.5 S = 13/2 (0.5+6.5) S = 13/2 ร 7 S = 91/2 S = 45.5 Now, Length of spiral = ๐ ร S = ๐ ร 45.5 = 22/7 ร 45.5 = 22 ร 6.5 = 143 โด Length of spiral is 143 cm
Ex 5.3
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