# 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 $\pi = \frac{22}{7}$)[Hint: Length of successive semicircles is $l_1 AcademicMathematicsNCERTClass 10 #### Complete Python Prime Pack for 2023 9 Courses 2 eBooks #### Artificial Intelligence & Machine Learning Prime Pack 6 Courses 1 eBooks #### Java Prime Pack 2023 9 Courses 2 eBooks Given: A spiral is made up of successive semicircles, with centres alternately at A and B, starting with centre at A, of radii 0.5 cm, 1.0 cm, 1.5 cm, 2.0 cm,… To do: We have to find the total length of such a spiral made up of thirteen consecutive semicircles. Solution: Here,$R_{1}=0.5\ cm, R_{2}=1.0\ cm, R_{3}=1.5\ cma=0.5\ cm, d=1.0\ cm-0.5\ cm=0.5\ cm$Length of the spiral$=$Sum of perimeter of$13$consecutive semicircles$=\pi R_{1}+\pi R_{2}+\pi R_{3}+\ldots . .+\pi R_{13}=\pi[R_{1}+R_{2}+R_{3}+\ldots . .R_{13}]=\pi[0.5+1.0+1.5+\ldots .+6.5]=\pi[\frac{13}{2}[2 \times 0.5+(13-1)(0.5)]]=\pi[\frac{13}{2}(1+12 \times 0.5)]=\pi \times(\frac{13}{2} \times 7)=\frac{22}{7} \times \frac{13}{2} \times 7=143\ cm\$

The total length of the spiral made up of thirteen consecutive semicircles is 143 cm.

Updated on 10-Oct-2022 13:20:30