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Question

A spiral is made up of successive semicircles, with centers alternatively at A and B, starting with center at A, of radii 0.5 cm, 1.0 cm, 1.5 cm, 2.0 cm, ... The total length of such a spiral made up of thirteen consecutive semicircles is ___ ( Round off to the nearest 10)

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Solution

Length of semi-circle =circumference of circle2 = 2πr2 = πr .

Length of semi-circle of radii 0.5 cm = π×0.5 cm

Length of semi-circle of radii 1.0 cm = π×1.0 cm

Length of semi-circle of radii 1.5 cm = π×1.5 cm

Length of semi-circle of radii 2.0 cm = π×2.0 cm

.....and so on

π(0.5),π(1.0),π(1.5)..... 13 term (There are total of thirteen semi-circles).
For total length of the spiral, we need to find sum of the sequence 13 terms
Total length of spiral = 0.5π+π+1.5π+2π........upto 13 terms

Total length of spiral = π(0.5+1.0+1.5+.......) upto 13 terms

Sequence 0.5, 1.0, 1.5 ....13 terms is an arithmetic progression.

Let's find the sum of this sequence.

a=0.5;d=0.5

S=n2(2a+(n1)d)

S=132(2×0.5+(131)(0.5))

S=132(1+6)

S=912=45.5

So

Total length of spiral = π(0.5+1.0+1.5+.......)

Total length of spiral = π(45.5)=143 cm


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