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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 cm, 1.0 cm, 1.5 cm, 2.0 cm, . . .as shown in Fig. 5.4. What is the total length of such a spiral made up of thirteen consecutive semicircles? [Take π = 22/7]

**Solution:**

To find the total length of the spiral, we will find the sum of first n terms, if they are in AP

The sum of the first n terms of an Arithmetic Progression is given by Sₙ = n/2 [2a + (n - 1) d]

Here, a is the first term, d is the common difference and n is the number of terms.

The Semiperimeter of a circle, l = πr

Given that the radii of the spiral follows a pattern of 0.5 cm, 1.0 cm, 1.5 cm, 2.0 cm,...

l₁ = π × (0.5 cm) = 0.5π cm [Since, radius = 0.5 cm]

l₂ = π × (1 cm) = π cm [Since, radius = 1 cm]

l₃ = π × (1.5 cm) = 1.5π cm [Since, radius = 1.5 cm]

Therefore, l₁, l₂, l₃, ... the lengths of the semi-circles are in an A.P

So,the AP is 0.5π, π, 1.5π, 2π, ...

a = 0.5π

d = π - 0.5π = 0.5π

We know that the sum of n terms of an A.P. is given by

Sₙ = n/2 [2a + (n - 1) d]

We need to calculate the total length of such a spiral made up of thirteen consecutive semicircles

Thus, n = 13

S₁₃ = 13/2 [2 × (0.5π) + (13 - 1)(0.5π) ]

= 13/2 [ π + 6π]

= 13/2 × 7π

= 13/2 × 7 × 22/7 (By taking π = 22/ 7)

= 143

Therefore, the length of such a spiral made up of thirteen consecutive semi-circles will be 143 cm.

**☛ Check: **NCERT Solutions for Class 10 Maths Chapter 5

**Video Solution:**

## A spiral is made up of successive semicircles, with centres alternately at A and B, starting with centre at A of radii 0.5, 1.0 cm, 1.5 cm, 2.0 cm, . . .as shown in Fig. 5.4. What is the total length of such a spiral made up of thirteen consecutive semicircles? [Take π = 22/7]

Class 10 Maths NCERT Solutions Chapter 5 Exercise 5.3 Question 18

**Summary:**

A spiral is made up of successive semicircles, with centres alternately at A and B, starting with centre at A of radii 0.5, 1.0 cm, 1.5 cm, 2.0 cm as shown in the figure, then the length of such a spiral made up of thirteen consecutive semicircles will be 143 cm.

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