Question

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 Fig. 5.4. What is the total length of such a spiral made up of thirteen consecutive semicircles? (Take \(\pi = \frac {22}{7}\))

Fig. 5.4

Hint:

Length of successive semicircles is \(l_1, l_2, l_3, l_4, ......\) with centres at \(A, B, A, B, ....\), respectively.

Answer 1

The semi-perimeter of a circle is \( \pi r \). The lengths of the first three semi-circles are calculated as follows: \( l_1 = \pi (0.5) = \frac {\pi}{2} \ cm \), \( l_2 = \pi (1) = \pi\ cm \), and \( l_3 = \pi (1.5) = \frac {3\pi}{2} cm \). These lengths, \( l_1, l_2, l_3 \), which represent the lengths of the semi-circles, form an Arithmetic Progression (A.P.): \( \frac \pi2, \pi, \frac {3\pi}{2}, 2\pi, ......\). The first term of this A.P. is \( a = \frac \pi2 \), and the common difference is \( d = \pi - \frac \pi2 = \frac \pi2 \). We need to find \( S_{13} \). The sum of n terms of an A.P. is given by the formula \( S_n = \frac n2[2a + (n-1)d] \).

Substituting the values for \( S_{13} \): \( S_{13} = \frac {13}{2}[2(\frac \pi2) + (13-1)(\frac \pi2)] \).

Simplifying the expression: \( S_{13} = \frac {13}{2}[\pi + \frac {12\pi}{2}] \).

Further simplification: \( S_{13} = (\frac{13}{2})(7\pi) \).

Resulting in: \( S_{13} = \frac {91\pi}{2} \). Using the approximation \( \pi \approx \frac{22}{7} \): \( S_{13} = 91 \times \frac {22}{2} \times 7 \). This calculation simplifies to \( S_{13} = 13 \times 11 \), which equals \( 143 \).

Therefore, the total length of a spiral composed of thirteen consecutive semi-circles is \( 143 \ cm \).