Mathematics
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 the fig. below. What is the total length of such a spiral made up of thirteen consecutive semicircles?
AP
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Answer
Circumference of semicircle = πr
Circumference of first circle = 0.5 π
Circumference of second circle = 1π = π
Circumference of third circle = 1.5π
Total length = 0.5π + π + 1.5π ……… upto 13 terms
The above list is an A.P. with first term (a) = 0.5π and common difference (d) = π - 0.5π = 0.5π
By formula,
Sum upto n terms = Sn =
Substituting values we get :
Hence, total length of spiral = 143 cm.
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