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
