Chapter 8.3, Problem 106E

To determine

To find: the total area of the shaded region.

Answer to Problem 106E

The area of shaded region is,

  $8.037$

  ${\text{inches}}^{\text{2}}$ .

Explanation of Solution

Given information:

The sides of a square are 27 inches in length.

Given figure

  

Calculation:

After the first division the area of the center square will be ${\left(\frac{1}{9}\right)}^{th}$ of the original square.

Therefore, the area of the center square is,

  $\frac{27}{9}$

Similarly, after division the area of the center squares is

  $8\times \frac{27}{9^2}$

Similarly, it will go on.

Therefore, the series is,

  $\frac{27}{9},27\times \frac{8}{9^2},27\times \frac{8^2}{9^3},\mathrm{...}$

The sum of this series is,

  $S_n=\frac{a_1\left(1-r^n\right)}{1-r}$

We know that,

  $\begin{array}{c}{a}_1=\frac{27}{9}\\ =3\\ r=\frac{8}{9}\end{array}$

Substituting,

  $\begin{array}{c}{S}_3=\frac{3\left(1-{\left(\frac{8}{9}\right)}^3\right)}{1-\left(\frac{8}{9}\right)}\\ =8.037\end{array}$

Therefore, the area of shaded region is,

  $8.037$ ${\text{inches}}^{\text{2}}$