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

  1. Chapter 5 Class 10 Arithmetic Progressions
  2. Serial order wise
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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 𝜋 = 22/7) Here we have to find total length, i.e., total circumference of all semicircles We know that Circumference of circle = 2πr So, circumference of semicircle = 1/2 × 2πr = πr Circumference of 1st semi circle = π × 0.5 Circumference of 2nd semicircle = π × 1.0 Circumference of 3rd semicircle = π × 1.5 …. Circumference of 13th semicircle = π × 6.5 So, the series is π × 0.5, π × 1.0, π × 1.5, ……………., π × 6.5 Taking 𝜋 common = 𝜋 (0.5, 1.0, 1. 5, 2 …….6.5) = 𝜋 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 = 𝑛/2 (𝑎+𝑙) Here n = 13, a = 0.5, 𝑙 = 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 = 𝜋 ×S = 𝜋 ×45.5 = 22/7×45.5 = 22 ×6.5 = 143 ∴ Length of spiral is 143 cm

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