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