Case Study I

A spiral is made up of successive semicircles, with centers alternately at A and B, starting with center at A, of radii 0.5 cm, 1 cm, 1.5 cm, 2 cm, …… as shown in the given figure.

Based on this information, answer the following questions:

(Take π = $\dfrac{22}{7}$)

1. What is the radius of the 9th semicircle?
   (a) 4 cm
   (b) 4.5 cm
   (c) 3.5 cm
   (d) 5 cm

2. The length of the 15th semicircle is :
   (a) 8π cm
   (b) 7π cm
   (c) 7.5π cm
   (d) 6.5π cm

3. The difference of lengths of the 19th and 12th semicircles is :
   (a) 3.5π cm
   (b) 3 cm
   (c) 4π cm
   (d) 4.5π cm

4. The total length of the spiral made up of first 7 consecutive semicircles is:
   (a) 38 cm
   (b) 42 cm
   (c) 44 cm
   (d) 46 cm

5. The lengths of the semicircles form an A.P. What is the common difference of this A.P.?
   (a) 0.5π cm
   (b) π cm
   (c) 0.25π cm
   (d) 1.25π cm

1. 0.5 cm, 1.0 cm, 1.5 cm, 2.0 cm, ….

The radius are in A.P. with

a = 0.5 cm

d = 1.0 - 0.5 = 0.5 cm

n = 9

We know that,

⇒ a_n = a + (n - 1)d

⇒ r₉ = 0.5 + (9 - 1)0.5

= 0.5 + (8)0.5

= 0.5 + 4

= 4.5 cm

Hence, option (b) is the correct option.

2. Calculating the radius of 15th semicircle.

We know that,

⇒ a_n = a + (n - 1)d

⇒ r₁₅ = 0.5 + (15 - 1)0.5

= 0.5 + 14(0.5)

= 0.5 + 7

= 7.5 cm

By formula,

Length of a semicircle = πr

= 7.5π cm

Hence, option (c) is the correct option.

3. We know that,

Length of a semicircle = πr

The lengths are: 0.5π cm, 1.0π cm, 1.5π cm, 2.0π cm,…. in an A.P. with,

a = 0.5π cm

d = 1.0π cm - 0.5π cm = 0.5π

The difference of lengths of the 19th and 12th semicircles is:

L₁₉ - L₁₂ = πr₁₉ - πr₁₂

= π(r₁₉ - r₁₂)

We know that,

a_n = a + (n - 1)d

⇒ r₁₉ = 0.5π + (19 - 1)0.5π

= 0.5π + 18(0.5π)

= 0.5π + 9π

= 9.5π cm.

⇒ r₁₂ = 0.5π + (12 - 1)0.5π

= 0.5π + 11(0.5π)

= 0.5π + 5.5π

= 6π cm.

L₁₉ - L₁₂ = (9.5π - 6π)

= 3.5π cm.

Hence, option (a) is the correct option.

4. The lengths are: 0.5π cm, 1.0π cm, 1.5π cm, 2.0π cm,….in an A.P.

a = 0.5π cm

d = 1.0π cm - 0.5π cm = 0.5π

The total length is the sum of the first 7 lengths S₇.

We know that,

⇒ S_n = $\dfrac{n}{2}$ [2a + (n - 1)d]

⇒ S₇ = $\dfrac{7}{2}$ [2(0.5π) + (7 - 1)(0.5π)]

= 3.5[1π + 6(0.5π)]

= 3.5[1π + 3π]

= 3.5(4π)

= 14π cm.

Total length = 14π cm

= $14 \times \dfrac{22}{7}$

= 44 cm.

Hence, option (c) is the correct option.

5. 0.5π cm, 1.0π cm, 1.5π cm, 2.0π cm, ….

d = 1.0π + 0.5π = 0.5 π cm.

Common difference of the A.P. of lengths = 0.5π cm.

Hence, option (a) is the correct option.