CBSE NCERT Solutions Chapter 5 Arithmetic Progressions Exercise 5.3 Question 18
18. A spiral is made up of successive semicircles, with centers alternatively at A and B, starting with center at A, of radii 0.5 cm, 1.0 cm, 1.5 cm, 2.0 cm, … What is the total length of such a spiral made up of thirteen consecutive semicircles.
Solution:
Length of semi-circle = Circumference of circle/2 =
=
Length of semi circle of radii 0.5 cm =
cm
Length of semi circle of radii 1.0 cm =
cm
Length of semi circle of radii 1.5 cm =
cm
Therefore, we have sequence of the form:
…… 13 terms {There are total of thirteen semi-circles}.
To find total length of the spiral, we need to find sum of the sequence
…… 13 terms
Total length of spiral
…… 13 terms
Total length of spiral =
…. 13 terms (1)
Sequence 0.5, 1.0, 1.5 ….13 terms is an arithmetic progression.
Lets find the sum of this sequence.
First term = a= 0.5
Common difference = 1.0 – 0.5 = 0.5
n = 13
Applying formula,
to find sum of n terms of AP , we get
Therefore, 0.5 + 1.0 + 1.5 + 2.0 ….13 terms = 45.5
Putting this in equation (1), we get
Total length of spiral =
) terms =
= 143 cm
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