Tiling a Patio Around a Fountain
A landscaper is tiling a square patio of side 8 m. In the center sits a circular fountain with diameter 3 m. The tiling covers the patio minus the fountain's footprint. Tiles are square, 0.5 m on a side, sold in boxes of 12. The landscaper must cut tiles around the circular edge, and experience shows that 10% of the tiles are wasted to cuts and breakage. The client asks whether a border of decorative edging (sold by the linear meter) is needed around the fountain; edging follows the circle's circumference. Use π ≈ 3.14.
1. Compute the area to be tiled and the number of boxes required, accounting for 10% waste.
Read the worked answer
Patio area = 8 × 8 = 64 m². Fountain radius = 1.5 m, area = πr² ≈ 3.14 × 2.25 = 7.065 m². Tiled area ≈ 64 − 7.065 = 56.935 m². Each tile covers 0.25 m², so tiles needed = 56.935 ÷ 0.25 ≈ 227.74, round up to 228 tiles. With 10% waste: 228 × 1.10 = 250.8, round up to 251 tiles. Boxes: 251 ÷ 12 ≈ 20.9, so 21 boxes (252 tiles).
2. How many linear meters of edging are needed for the fountain, and is the estimate sensitive to using π ≈ 3.14 versus 3.1416?
Read the worked answer
Circumference = πd = 3.14 × 3 = 9.42 m. With π = 3.1416: 9.4248 m. The difference is about 0.005 m (5 mm), negligible for ordering; ordering 9.5 m gives a small margin for joins.
3. The client wants the tiled area to be at least 57 m². Does the current fountain size allow this? What is the largest fountain diameter (to 0.1 m) that keeps tiled area ≥ 57 m²?
Read the worked answer
Current tiled area ≈ 56.935 m², which is just under 57 m², so it fails by about 0.065 m². Need 64 − πr² ≥ 57 → πr² ≤ 7 → r² ≤ 7 ÷ 3.14 ≈ 2.229 → r ≈ 1.493 m, diameter ≈ 2.986 m, so the largest diameter to 0.1 m is 2.9 m. (At 3.0 m the area is 64 − 7.065 = 56.935 < 57.)