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

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,

$R_{1}=0.5\ cm, R_{2}=1.0\ cm, R_{3}=1.5\ cm$

$a=0.5\ cm, d=1.0\ cm-0.5\ cm=0.5\ cm$

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.

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

Advertisements