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

  1. Chapter 5 Class 10 Arithmetic Progressions
  2. Serial order wise

Transcript

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, โ€ฆโ€ฆโ€ฆ as shown in figure. What is the total length of such a spiral made up of thirteen consecutive semicircles? (take ๐œ‹ = 22/7) Here we have to find total length, i.e., total circumference of all semicircles We know that Circumference of circle = 2ฯ€r So, Circumference of semicircle = 1/2 ร— 2ฯ€r = ฯ€r 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 โ€ฆ. Circumference of 13th semicircle = ฯ€ ร— 6.5 So, the series is ฯ€ ร— 0.5, ฯ€ ร— 1.0, ฯ€ ร— 1.5, โ€ฆโ€ฆโ€ฆโ€ฆโ€ฆ., ฯ€ ร— 6.5 Taking ๐œ‹ common = ๐œ‹ (0.5, 1.0, 1. 5, 2 โ€ฆโ€ฆ.6.5) = ๐… S Where S is sum of series 0.5, 1.0, 1.5, 2 โ€ฆโ€ฆ.6.5 For 0.5, 1.0, 1. 5, 2 โ€ฆโ€ฆ.6.5 As difference is same, it is an AP S can be found by using formula S = ๐’/๐Ÿ (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œ‹ ร— S = ๐… ร— 45.5 = 22/7 ร— 45.5 = 22 ร— 6.5 = 143 โˆด Length of spiral is 143 cm

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Davneet Singh
Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 10 years. He provides courses for Maths and Science at Teachoo.