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`)

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

Answer 1

Length of a semi-circle

= semi circumference = `1/2 (2pir) = pir`

_l1 = πr₁ = 0.5 π cm = 1 × 0.5 π cm

_l2 = πr₂ = 1.0 π cm = 2 × 0.5 π cm

_l3 = πr₃ = 1.5 π cm = 3 × 0.5 π cm 

_l4 = πr₄ = 2.0 c cm = 4 × 0.5 π cm

_l13 = πr₁₃ = 13 × 0.5 π cm = 6.5 π cm

Now, length of the spiral = l₁ + l₂ + l₃ + l₄ + ... + l₁₃

= 0.5 π [1 + 2 + 3 + 4 + ... + 13] cm     ....(1)

∴ 1, 2, 3, 4, ...., 13 are in AP such that

a = l and l = 13

∴ `S_13 = 13/2 [1 + 13]       ....["Using" S_n = n/2 (a + l)]`

= `13/2 xx 14`

= 13 × 7

= 91

∴ From (1), we have:

Total length of the spiral = 0.5 π [91] cm

= `5/10 xx 22/7 xx 91` cm

= 11 × 13 cm

= 143 cm