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 = ) 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

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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, ⇒ an = 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, ⇒ an = 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, an = 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, ⇒ Sn = [2a + (n - 1)d] ⇒ S7 = [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 = = 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.

Mathematics

AP

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Mathematics

AP

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