Question

The area of the entire patio is
2a2-4a+5cm2. The area of the square fountain in the center is a2-2a-6cm2. What is the area of the tiled region?

Answer 1

To find the area of the tiled region surrounding the square fountain, we need to subtract the area of the fountain from the area of the entire patio.

1. **Area of the entire patio:**
\[
A_{\text{patio}} = 2a^2 - 4a + 5 \text{ cm}^2
\]

2. **Area of the square fountain:**
\[
A_{\text{fountain}} = a^2 - 2a - 6 \text{ cm}^2
\]

3. **Area of the tiled region:**
To find the area of the tiled region, we subtract the area of the fountain from the area of the patio:
\[
A_{\text{tiled}} = A_{\text{patio}} - A_{\text{fountain}}
\]

Plugging in the values:
\[
A_{\text{tiled}} = (2a^2 - 4a + 5) - (a^2 - 2a - 6)
\]

4. **Simplifying the expression:**
Distribute the negative sign in front of the second area:
\[
A_{\text{tiled}} = 2a^2 - 4a + 5 - a^2 + 2a + 6
\]

Combine like terms:
\[
A_{\text{tiled}} = (2a^2 - a^2) + (-4a + 2a) + (5 + 6)
\]
\[
A_{\text{tiled}} = a^2 - 2a + 11
\]

Thus, the area of the tiled region is:
\[
\boxed{a^2 - 2a + 11 \text{ cm}^2}
\]