Part 1. The area of a circular pool is approximately 7,850 square feet. The owner wants to replace the tiling at the edge of the pool. The edging is 6 inches wide, so he plans to use 6-inch square tiles to form a continuous inner edge. How many tiles will he need to purchase? | Part 2. Once the square tiles are in place around the pool, there will be extra space between the tiles. What shape of tile will best fill this space? How many tiles of this shape should he purchase?

Part 1. To calculate the number of tiles needed to form a continuous inner edge around the circular pool, we need to determine the circumference of the pool and then divide it by the perimeter of each tile.

Step 1: Find the radius of the pool
The formula for the area of a circle is A = πr^2, where A is the area and r is the radius. Rearranging the formula, we can find the radius: r = √(A/π).

Given the area is approximately 7,850 square feet, we can calculate the radius:
r = √(7850/π) ≈ 50 feet.

Step 2: Calculate the circumference of the pool
The circumference of a circle is given by the formula C = 2πr.

Taking the radius as 50 feet, we can calculate the circumference:
C = 2π(50) ≈ 314.16 feet.

Step 3: Determine the number of tiles needed
Since the edging is 6 inches wide, or 0.5 feet, we can calculate the number of tiles required by dividing the circumference by the perimeter of each tile.

The perimeter of a 6-inch square tile is 4 * 0.5 feet = 2 feet.

Number of tiles needed = Circumference / Tile perimeter
= 314.16 ft / 2 ft
= 157.08 tiles.

Therefore, the owner will need to purchase approximately 157 tiles to replace the tiling at the edge of the pool.

Part 2. Once the square tiles are in place, there will be extra space between them due to the circular shape of the pool. To fill this space, the owner should choose a tile shape that best fits the gaps.

For circular pool edges, it is best to use circular or curved tiles to fill the space between the square tiles. Circular tiles, such as quarter-circle or half-circle tiles, can be used to create a seamless transition between the square tiles and fill the gaps more effectively.

To determine the number of circular tiles needed, we need to calculate the length of the gaps around the pool and divide it by the circumference of each circular tile.

Step 1: Calculate the length of the gaps
The length of the gaps can be determined by subtracting the circumference of the pool from the inner circumference formed by the square tiles.

Inner circumference = Circumference of the pool + (2 * Tile width)
= 314.16 ft + (2 * 0.5 ft)
= 315.16 ft.

Length of gaps = Inner circumference - Circumference of the pool
= 315.16 ft - 314.16 ft
= 1 ft.

Step 2: Determine the number of circular tiles needed
To calculate the number of circular tiles required, we divide the length of the gaps by the circumference of each circular tile.

Assuming the chosen circular tile has a circumference of 1 foot, we can calculate:
Number of circular tiles needed = Length of gaps / Circumference of each tile
= 1 ft / 1 ft
= 1 tile.

Therefore, the owner will need to purchase 1 circular tile to fill the space between the square tiles around the pool.