Chapter 9.3, Problem 91E

To determine

To find: The total area of the shaded region.

Answer to Problem 91E

  $273\frac{8}{9}{\text{in}}^2$

Explanation of Solution

Given: The sides of a square are 27 inches in length. New squares are formed by dividing the original square into nine squares. The center square is then shaded. If this process is repeated three more times.

  

The original length of square is 27 inches.

  

First divide into 9 squares.

So, the length of side of each square be 9 inches because each sides divides by three lines.

Now shade the center square.

  

Area of shaded square in step 1 $=9\times 9=81{\text{in}}^2$

Now, total 8 white squares and repeat the process. The length of each square is 9 inches.

Now, divide each square into 9 equal squares and shaded the center square.

  

The length of each square will be 3 inches.

Area of shaded square in step 2 $=8\times 3\times 3=72{\text{in}}^2$

Now, total 64 white squares and repeat the process. The length of each square is 3 inches.

Now, divide each square into 9 equal squares and shaded the center square.

  

The length of each square will be 1 inches.

Area of shaded square in step 3 $=64\times 1\times 1=64{\text{in}}^2$

Now, total 512 white squares and repeat the process. The length of each square is 1 inches.

Now, divide each square into 9 equal squares and shaded the center square. The length of each square will be $\frac{1}{3}$ inches.

Area of shaded squares in step 4 $=512\times \frac{1}{3}\times \frac{1}{3}=\frac{512}{9}{\text{in}}^2$

Total area of the shaded squares $=81+72+64+\frac{512}{9}=273\frac{8}{9}{\text{in}}^2$