Class 11

Math

Algebra

Sequences and Series

A spiral is made up of successive semicircles, with centers alternately at A and B, starting wit center at A. of radii $0.5cm,1.0cm,1.5cm,2.0cm,.....$ as shown if 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 $L_{1}, L_{2},L_{3},.....$ with centers at A, B, A, B,....., respectively, }

$r_{1}=0.5cm⇒(1×0.5)$

$r_{2}=1.0cm⇒(2×0.5)$

$r_{3}=1.5cm⇒(3×0.5)$

$r_{4}=2.0cm⇒(4×0.5)$

The radii are in AP

$a_{1}=0.5,d=0.5,r_{13}=13×0.5=6.5cm$

Total length of the spiral$=l_{1}+l_{2}+l_{3}+..............+l_{13}$

Circumference of a semicircle$=πr$

$∴$ Total lenght of the spiral$=π_{1}×0.5+π×1.0+......+π×6.5$

$=π×21 (1+2+3+........+13)$

$[∴S_{n}=2n(n+1) ]$

$=722 ×21 ×213×14 =11×13=143$

$∴$ Total length of spiral$=143cm$