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A spiral is made up of successive semicircles, with centers al

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A spiral is made up of successive semicircles, with centers alternately at $A$ and $B$ , starting with center at $A$ , of radii $0.5 c m, 1.0 c m, 1.5 c m, 2.0 c m$ , . . . as shown in Fig. What is the total length of such a spiral made up of thirteen consecutive semicircles

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A spiral is made up of successive semicircles, with centers alternately at

A

and

B

, starting with center at

A

, of radii

0.5 c m, 1.0 c m, 1.5 c m, 2.0 c m

, . . . as shown in Fig. What is the total length of such a spiral made up of thirteen consecutive semicircles

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Text solutionVerified

Circumference of first semicircle

=

π r = 0.5 π

Circumference of second semicircle

=

π r = π

Circumference of third semicircle

=

π r = 1.5 π

It is clear that

a = 0.5 π

d = 0.5 π

and

n = 13

Hence, length of spiral can be calculated as follows:

S = \frac{n}{2} [2 a + (n - 1) d]

=

\frac{13}{2} (2 \times 0.5 π + 12 \times 0.5 π)

=

\frac{13}{2} \times 7 π

=

\frac{13}{2} \times 7 \times \frac{22}{7}

= 143

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Question Text	A spiral is made up of successive semicircles, with centers alternately at $A$ and $B$ , starting with center at $A$ , of radii $0.5 c m, 1.0 c m, 1.5 c m, 2.0 c m$ , . . . as shown in Fig. What is the total length of such a spiral made up of thirteen consecutive semicircles
Answer Type	Text solution:1
Upvotes	150