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.
☛ Related Questions:
- Find the sum of the odd numbers between 0 and 50.
- A contract on a construction job specifies a penalty for delay of completion beyond a certain date as follows: Rs. 200 for the first day, Rs. 250 for the second day, Rs. 300 for the third day, etc., the penalty for each succeeding day being Rs. 50 more than for the preceding day. How much money the contractor has to pay as penalty if he has delayed the work by 30 days?
- A sum of Rs 700 is to be used to give seven cash prizes to students of a school for their overall academic performance. If each prize is Rs 20 less than its preceding prize, find the value of each of the prizes.
- In a school, students thought of planting trees in and around the school to reduce air pollution. It was decided that the number of trees, that each section of each class will plant, will be the same as the class, in which they are studying, e.g., a section of Class I will plant 1 tree, a section of Class II will plant 2 trees and so on till Class XII. There are three sections of each class. How many trees will be planted by the students?