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

Ex 5.3, 18 - Chapter 5 Class 10 Arithmetic Progressions - Part 2
Ex 5.3, 18 - Chapter 5 Class 10 Arithmetic Progressions - Part 3 Ex 5.3, 18 - Chapter 5 Class 10 Arithmetic Progressions - Part 4

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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œ‹ = 22/7) Here we have to find total length, i.e., total circumference of all semicircles We know that Circumference of circle = 2Ο€r So, Circumference of semicircle = 1/2 Γ— 2Ο€r = Ο€r Circumference of 1st semi circle = Ο€ Γ— 0.5 Circumference of 2nd semicircle = Ο€ Γ— 1.0 Circumference of 3rd semicircle = Ο€ Γ— 1.5 …. Circumference of 13th semicircle = Ο€ Γ— 6.5 So, the series is Ο€ Γ— 0.5, Ο€ Γ— 1.0, Ο€ Γ— 1.5, ……………., Ο€ Γ— 6.5 Taking πœ‹ common = πœ‹ (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) Putting n = 13, a = 0.5, 𝑙 = 6.5 S = 13/2 (0.5+6.5) S = 13/2 Γ— 7 S = 91/2 S = 45.5 Now, Length of spiral = πœ‹ Γ— S = 𝝅 Γ— 45.5 = 22/7 Γ— 45.5 = 22 Γ— 6.5 = 143 ∴ Length of spiral is 143 cm

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Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 13 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.