Question

swimming pool. each square tile has an area of n. write an expression showing how many tiles it will take to complete a border around the outside of the pool. (note: dont forget the corners.)

Explanation:

Step1: Analyze the pool's dimensions

From the diagram, the pool is a \( 5 \times 5 \) grid of tiles (assuming each small square is a tile unit for the pool). To add a border, we need to consider the outer layer. The new length and width of the pool with the border will be \( 5 + 2 = 7 \) (adding 1 tile on each side). But wait, actually, for a border around a rectangle (or square) of size \( n \times n \) (here \( n = 5 \)), the number of border tiles can be calculated by considering the perimeter and the corner tiles, but a better way is:

The total number of tiles including the border is \( (n + 2) \times (n + 2) \) (since we add 1 tile on each side, so length and width increase by 2), and the number of tiles in the pool is \( n \times n \). So the number of border tiles is \( (n + 2)^2 - n^2 \).

Wait, but let's check with \( n = 5 \). The pool is \( 5 \times 5 \), so pool tiles: \( 5 \times 5 = 25 \). The bordered area is \( (5 + 2) \times (5 + 2)= 49 \). So border tiles: \( 49 - 25 = 24 \). Alternatively, we can calculate the perimeter - 4 (to avoid double - counting corners) plus 4 corners? Wait no, the formula \( (n + 2)^2 - n^2 \) expands to \( n^2 + 4n + 4 - n^2 = 4n + 4 \). For \( n = 5 \), \( 4\times5 + 4 = 24 \), which matches.

Wait, but the problem says "each square tile has an area of \( n \)"? Wait, maybe I misread. Wait the original problem: "swimming pool. Each square tile has an area of \( n \). Write an expression showing how many tiles it will take to complete a border around the outside of the pool. (Note: Don’t forget the corners.)" Wait, maybe the pool is a square with side length \( s \) (in terms of tiles), and each tile has area \( n \). But from the diagram, the pool is \( 5 \) tiles per side (since the grid is 5x5). Let's assume the pool is a square with side length \( s \) (number of tiles in the pool's side). Then, to add a border, we need to calculate the number of tiles in the border.

Another approach: For a square pool with side length \( s \) (number of tiles), the border tiles can be calculated as follows:

- The top and bottom borders: each has \( s + 2 \) tiles (because we add one tile on each end), so \( 2\times(s + 2) \) tiles.
- The left and right borders: but we have already counted the corner tiles in the top and bottom, so each of these has \( s \) tiles (excluding the top and bottom corners), so \( 2\times s \) tiles.

Total border tiles: \( 2(s + 2)+2s=2s + 4+2s = 4s + 4=4(s + 1) \).

If the pool in the diagram is \( 5 \) tiles per side ( \( s = 5 \) ), then \( 4\times(5 + 1)=24 \), which matches the earlier calculation.

But the problem says "each square tile has an area of \( n \)". Wait, maybe the side length of the pool (in terms of tiles) is, say, \( s \), and we need to find the number of tiles for the border. Let's assume the pool is a square with side length \( s \) (number of tiles). Then the number of border tiles is \( 4(s + 1) \) (or \( 4s+4 \)).

Wait, maybe the pool is a square with side length \( 5 \) (from the diagram, 5x5 grid). Let's confirm with \( s = 5 \):

- Top border: \( 5 + 2 = 7 \) tiles (including left and right corners)
- Bottom border: \( 7 \) tiles
- Left border (excluding top and bottom corners): \( 5 \) tiles
- Right border (excluding top and bottom corners): \( 5 \) tiles
- Total: \( 7 + 7+5 + 5=24 \), and \( 4\times(5 + 1)=24 \), \( 4\times5+4 = 24 \).

So the general formula for a square pool with side length \( s \) (number of tiles) is \( 4(s + 1) \) or \( 4s + 4 \).

But the problem says "each square tile has an area of \( n \)". Wait, maybe there's a misinterpretation. Maybe the side length of the tile is related to \( n \), but the key is the number of tiles. Let's assume the pool is a square with side length \( s \) (in tiles), then the number of border tiles is \( 4(s + 1) \). If the pool in the diagram is \( 5 \) tiles per side, then \( s = 5 \), and the number of border tiles is \( 4\times(5 + 1)=24 \). But the problem says "write an expression", so if we let the side length of the pool (in tiles) be \( s \), the expression for the number of border tiles is \( 4(s + 1) \) or \( 4s + 4 \).

Alternatively, if we consider the area of each tile is \( n \), but we are counting the number of tiles, not the area. So the number of tiles for the border depends on the side length of the pool (in tiles).

Assuming the pool is a square with side length \( s \) (number of tiles), the expression for the number of border tiles is \( 4(s + 1) \) (or \( 4s + 4 \)).

Answer:

If the side length of the pool (in number of tiles) is \( s \), the expression for the number of border tiles is \( \boldsymbol{4(s + 1)} \) (or \( \boldsymbol{4s + 4} \)). If the pool in the diagram has a side length of \( 5 \) (as per the 5x5 grid), substituting \( s = 5 \) gives \( 4\times(5 + 1)=24 \) tiles.