   Chapter 11.2, Problem 92E

Chapter
Section
Textbook Problem

# In the figure at the right there are infinitely many circles approaching the vertices of an equilateral triangle, each circle touching other circles and sides of the triangle. If the triangle has sides of length 1, find the total area occupied by the circles.

To determine

To find: The total area occupied by the circles.

Explanation

Given:

The triangle has sides of length 1.

Proof:

Consider the radius of the large circle is r1 and radius of the next circle r2 and so on.

From the above Figure 1, BAC=60 .

cos60=r1|AB|12=r1|AB||AB|=2r1

From the Figure 1, DBC=60

cos60=r2|DB|12=r2|DB||DB|=2r2

2r1=r1+r2+2r22r1=r1+3r2r1=3r2

In general,

3rn+1=rnrn+1=13rn

The total area A is sum of the area of circles.

A=πr12+3πr22+3πr32+=πr12+3πr22+3π(13r2)2+3π(132r2)2=πr12+3πr22(1+13

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