Question

a decorative tile is in the shape of a square. it is made up of a smaller square of side length ( b ) centimeters which is surrounded by a border of width 4 centimeters. find an expression for the total area (in square centimeters) of the decorative tile in terms of ( b ). simplify the result.

Explanation:

Step1: Determine side length of total tile

The smaller square has side length \( b \), and the border has width 4 cm on both sides (top/bottom and left/right). So the total side length \( s \) of the decorative tile is \( b + 4 + 4 = b + 8 \).

Step2: Calculate area of total tile

The area \( A \) of a square is side length squared, so \( A=(b + 8)^2 \).

Step3: Expand the square

Using the formula \( (a + b)^2=a^2+2ab + b^2 \), where \( a = b \) and \( b = 8 \), we get \( (b + 8)^2=b^2+16b + 64 \).

Answer:

The total area of the decorative tile is \( \boldsymbol{b^2 + 16b + 64} \) square centimeters.