We have to find out the total length of such a spiral made up of thirteen consecutive semicircle.
We have learnt that
The perimeter of a semi-circle = πr
P1 = π(0.5) = π/2 cm
P2 = π(1) = π cm
P3 = π(1.5) = 3π/2 cm
Where, P1, P2, P3 are the lengths of the semi-circles.
Hence we get the series as
π/2, π, 3π/2, 2π, ….
P1 = π/2 cm
P2 = π cm
Common difference, d = P2 – P1 = π – π/2 = π/2
First term = P1= a = π/2 cm
We know that by sum of n term formula,
Sn = n/2 [2a + (n – 1)d]
Thereforee, Sum of the length of 13 consecutive circles is;
S13 = 13/2 [2(π/2) + (13 – 1)π/2]
= 13/2 [π + 6π]
= 13/2 × 7 × 22/7
= 143 cm
The total length of such a spiral made up of thirteen consecutive semicircles= 143cm