Chapter 8.2, Problem 68E

To determine

To Find: The total area occupied by the circles.

Answer to Problem 68E

The area of the circle is occupied by the circles is $\frac{11\pi }{96}$ .

Explanation of Solution

Given:

The given diagram is shown in Figure 1

  

Figure 1

Calculation:

Consider the area of the equilateral triangle is,

  $A=\frac{\sqrt{3}}{4}{a}^2$

Then, for the side 1 the area is,

  $\begin{array}{c}A=\frac{\sqrt{3}}{4}{\left(1\right)}^2\\ =\frac{\sqrt{3}}{4}\end{array}$

Then, the radius of the central inscribed circle is $\frac{\sqrt{3}}{6}$ as the area of the first circle as,

  $A=\pi {\left(\frac{\sqrt{3}}{6}\right)}^2$

Then, the area of the infinite circle is,

  $\begin{array}{c}A=A_1+A_2+A_3\mathrm{....}+A_n\\ =\frac{\pi }{12}+3\cdot \frac{\pi }{12}\cdot \frac{1}{9}+3\cdot \frac{\pi }{12}\cdot \frac{1}{9^2}+3\cdot \frac{\pi }{12}\cdot \frac{1}{9^{}}\\ =\frac{\pi }{12}+3\cdot \frac{\pi }{12}\pi \left(\frac{\frac{1}{9}}{1-\frac{1}{9}}\right)\\ =\frac{11\pi }{96}\end{array}$