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29 Dec 2021
Problem 68
Page 575
Section 8.2: Series
Chapter 8: Infinite Sequences and Series
Textbook ExpertVerified Tutor
29 Dec 2021
Given information
In the given figure there are infinitely many circles approaching the vertices of an equilateral triangle, each circle touching other circles and sides of the triangle. The length of each sides of the triangle is 
Step-by-step explanation
Step 1.
We are given that the triangle is equilateral, basic geometry tells us that the inscribed circle of such a triangle has it's center at 
where 
the height.
where the height.Also using Pythagora's Theorem we write, (where a is the length of a side of the triangle)
Next notice that every one of the little circles creates a new smaller triangle (look at the black line)
This new triangle has height $\frac{1}{3} h$ because it begins after the circle with 
Now because of similar triangles theorem the small one is also equilateral meaning the
radius of the small circle is now $\frac{1}{3}$ of the new height therefore
We can continue the same logic for all the smaller circles and their radius will be
Single Variable Calculus: Early Transcendentals
4th Edition, 2018
Stewart
ISBN: 9781337687805
Solutions
Chapter
Section
- Problem 1a
- Problem 1b
- Problem 2
- Problem 3
- Problem 4
- Problem 5
- Problem 6
- Problem 7
- Problem 8
- Problem 9a
- Problem 9b
- Problem 10a
- Problem 10b
- Problem 11
- Problem 12
- Problem 13
- Problem 14
- Problem 15
- Problem 16
- Problem 17
- Problem 18
- Problem 19
- Problem 20
- Problem 21
- Problem 22
- Problem 23
- Problem 24
- Problem 25
- Problem 26
- Problem 27
- Problem 28
- Problem 29
- Problem 30
- Problem 32
- Problem 32
- Problem 33
- Problem 34
- Problem 35a
- Problem 35b
- Problem 35c
- Problem 35d
- Problem 36
- Problem 37
- Problem 39
- Problem 40
- Problem 41
- Problem 41
- Problem 42
- Problem 43
- Problem 44
- Problem 45
- Problem 46
- Problem 47
- Problem 48
- Problem 49a
- Problem 49b
- Problem 49c
- Problem 50a
- Problem 50b
- Problem 51a
- Problem 51b
- Problem 52a
- Problem 52b
- Problem 52c
- Problem 53
- Problem 54
- Problem 55
- Problem 56
- Problem 57
- Problem 59
- Problem 60
- Problem 61
- Problem 62
- Problem 63
- Problem 64a
- Problem 64b
- Problem 64c
- Problem 65a
- Problem 65b
- Problem 66a
- Problem 66b
- Problem 67a
- Problem 67b
- Problem 67c
- Problem 68