Question

a spiral is made up of successive semicircles with centres alternately at A and B starting with centres at A of radius 0.5 CM ,1 cm 1.5 cm ,2cm what is the total length of such a spiral made up of 13 consecutive semicircle

Answer 1

Its the attachment of the answer...and the total final answer is 143. If it does help you please reply me☺️ Because I'm new to here

Answer 2

$\small\sf\red{Circumference\:of\:semi-circle\:=\frac{1}{2}(2πr)=πr}$

$\small\sf\blue{Radii\:of\:semi-circle=0.5cm,1.0cm,1.5cm,...}$

$\therefore$$\small\sf\green{The\:length\:of\:first\:spiral\:l1=π(0.5cm)}$

$\small\sf\orange{Similarly,the\:length\:of\:spiral\:l1,l2,l3,l4,...\:are\:as\:follows:}$

$\large\tt\purple{a=0.5π}$

$\large\tt\purple{d=1.0π-0.5π=0.5π}$

$\large\tt\purple{n=13}$

$\small\sf\underline\red{The\:sum\:of\:n\:terms\:of\:an\:AP\:is\:given\:by}$

$\large\tt\purple{Sn=\frac{n}{2}(2a+(n-1)d}$

$\longrightarrow$$\large\tt\purple{S13=\frac{13}{2}(2(0.5π)+(13-1)0.5π}$

$\longrightarrow$$\large\tt\purple{6.5(π+6π)}$

$\longrightarrow$$\large\tt\purple{6.5(7×\frac{22}{7})}$

$\longrightarrow$$\large\tt\purple{6.5×22=143cm}$

$\small\sf\underline\pink{Hence,the\:total\:length\:of\:spiral\:made\:up\:of\:13\:semicircles\:is\:143cm}$