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 Figure. What is the total length of such a spiral made up of thirteen consecutive semicircles? (Take `pi=22/7`)

Explanation

Let `L_(1), L_(2), L_(3), L_(4),…L_(13)` be the lengths of semicircles of radii 0.5cm, 1 cm, 1.5cm, 2 cm, … and `(13)/(2)` cm respectively.
Then, we have
`L_(1) = (pi xx 0.5) cm = (pi)/(2)cm,`
`L_(2) = (pi xx 1)cm = 2((pi)/(2))cm,`
`L_(3) = (pi xx 1.5)cm = 3((pi)/(2))cm,`
`L_(4) = (pi xx 2)cm = 4((pi)/(2))cm,`
... .... ... .... ...
... ... ... ... ... ...
`L_(13) = (pi xx (13)/(2))cm = 13((pi)/(2))cm.`
`therefore `total length of the spiral
`= L_(1) + L_(2) + L_(3) + L_(4) +... + L_(13)`
` = {(pi)/(2) + 2((pi)/(2)) + 3((pi)/(2)) + 4((pi)/(2)) + ... + 13((pi)/(2))}cm`
` = (pi)/(2)(1+2+3+4+..+13)cm`
` = (pi)/(2) xx (13)/(2) xx (1+13)cm " " [S_(n) = (n)/(2)(a+l)]`
` = ((1)/(2) xx (22)/(7) xx (13)/(2) xx 14)cm = 143cm`
Hence, the required length of the spiral is 143cm.