Question

A spiral is made up of successive semicircles, with centers alternatively at A and B, starting with center at A, of radii 0.5 cm, 1.0 cm, 1.5 cm, 2.0 cm, ... What is the total length of such a spiral made up of thirteen consecutive semicircles?

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Answer 1

Length of semi-circle =$\frac{circumferenceofcircle}{2}$ = $\frac{2\pi r}{2}$ = $\pi r$ .

Length of semi-circle of radii 0.5 cm = $\pi \times 0.5cm$

Length of semi-circle of radii 1.0 cm = $\pi \times 1.0cm$

Length of semi-circle of radii 1.5 cm = $\pi \times 1.5cm$

Length of semi-circle of radii 2.0 cm = $\pi \times 2.0cm$

.....and so on

$\pi \left(0.5\right),\pi \left(1.0\right),\pi \left(1.5\right).....$ 13 term (There are total of thirteen semi-circles}.
For total length of the spiral, we need to find sum of the sequence 13 terms
Total length of spiral = $0.5\pi +\pi +1.5\pi +2\pi ........upto13terms$

$\Rightarrow$ Total length of spiral = $\pi (0.5+1.0+1.5+.......)upto13terms$

Sequence 0.5, 1.0, 1.5 ....13 terms is an arithmetic progression.

Let's find the sum of this sequence.

$a=0.5;d=0.5$

$S=\frac{n}{2}(2a+(n-1\left)d\right)$

$\Rightarrow S=\frac{13}{2}(2\times 0.5+(13-1\left)\right(0.5\left)\right)$

$\Rightarrow S=\frac{13}{2}(1+6)$

$\Rightarrow S=\frac{91}{2}=45.5$

So

$\Rightarrow$ Total length of spiral = $\pi (0.5+1.0+1.5+.......)$

$\Rightarrow$ Total length of spiral = $\pi \left(45.5\right)=143cm$