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A spiral is made up of successive semicircles, with centres alternately at A and B, starting with centre at A, of radii $0.5\text{ cm}, 1.0\text{ cm}, 1.5\text{ cm}, 2.0\text{ cm}, \ldots$ as shown. What is the total length of the spiral made up of thirteen consecutive semicircles? Take $\pi=\frac{22}{7}$. 
A$66\text{ cm}$
B$71.5\text{ cm}$
C$78\text{ cm}$
D$84.5\text{ cm}$
Answer & Solution
Correct answer: B. $71.5\text{ cm}$
The length of each semicircle is $\pi r$. The radii form an AP: $0.5,1.0,1.5,\ldots$ with $n=13$, $a=0.5$, $d=0.5$. So the sum of radii is
$$S_{13}=\frac{13}{2}[2(0.5)+12(0.5)]=\frac{13}{2}(1+6)=\frac{91}{2}=45.5.$$
Hence total spiral length
$$=\pi \times 45.5=\frac{22}{7}\times 45.5=143.$$
This is the total length in cm.
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