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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? - CBSE Class 10 - Mathematics

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Question

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 `pi = 22/7` )

Solution

Semi-perimeter of circle = πr

I1 = π(0.5) = `pi/2 `

I2 = π(1) = π cm

I3 = π(1.5) = `(3pi)/2 `

Therefore, I1, I2, I3 ,i.e. the lengths of the semi-circles are in an A.P.,

`pi/2, pi, (3pi)/2, 2pi ........`

`a = pi/2`

`d = pi - pi/2=pi/2`

S13 =?

We know that the sum of n terms of an a A.P. is given by

`S_n = n/2[2a+(n-1)d]`

`= 13/2[2(pi/2)+(13-1)(pi/2)]`

`= 13/2[pi+(12pi)/2]`

`=(13/2)(7pi)`

`=(91pi)/2`

`= (91xx22)/2xx7 = 13xx11`

= 143

Therefore, the length of such spiral of thirteen consecutive semi-circles will be 143 cm

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 NCERT Solution for Mathematics Textbook for Class 10 (2019 to Current)
Chapter 5: Arithmetic Progressions
Ex. 5.30 | Q: 18 | Page no. 113
Solution 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? Concept: Sum of First n Terms of an AP.
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