(Q35).A spiral is made up of successive semicircles, with centres alternately at A and B, starting with centre at A, of radii 0.5 cm, 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 =
22
7
 )

[Hint: Length of successive semicircles is l1, l2, l3, l4,….. with centres at A, B, A, B,…, respectively.]

.AB

Given: l1, l2, l3, l4,….., l13 are the semicircles with centres alternately at A and B; with radii

r1 = 0.5 cm [ × 0.5]

r2 = 1.0 cm [ × 0.5]

r3 = 1.5 cm [ × 0.5]

r4 = 2.0 cm [ × 0.5] [∵ Radii are in A.P. as aj = 0.5 and d = 0.5]

..................

r13 = × 0.5 =

Now, the total length of the spiral = l1 + l2 + l3 + l4 + ….. + l13 [∵ 13 given]

But circumference of a semi-cirle is πr.

∴ Total length of the spiral = π × 0.5 + π × 1.0 + ………. + π × 6.5

= π ×
1
2
 [l + 2 + 3 + ….. + 13]
[∵ Sum of the first n – natural numbers is
n(n+1)
2
=
×
1
 ×
×
2

= ×

⇒ x = – =

= cm.