A spiral is made up of successive semicircles, with centers alternatively at A and B, starting with center at A first and then at B, with radii 0.5 cm, 1.0 cm, 1.5 cm, 2.0 cm and so on. The total length of such a spiral made up of 13 consecutive semi circles is
Length of a 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 the total length of the spiral, we need to find sum of the 13 terms.
Total length of the 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+(n−1)d)
⇒S=132(2×0.5+(13−1)(0.5))
⇒S=132(1+6)
⇒S=912=45.5
⇒ Total length of spiral = π(0.5+1.0+1.5+.......)
⇒ Total length of spiral = π(45.5)=143 cm