Courtyard Tiling, Fountain Waste, and a Fixed Budget
The student council wants to tile a rectangular courtyard that measures 12.6 m by 8.4 m. A circular fountain with radius 1.5 m stands in the middle and will not be tiled. The chosen square tiles are 0.30 m on a side and are sold only in boxes of 25. The supplier recommends allowing 8% extra tiles for cuts and breakage; the council's maximum budget is 1,500 BAM, and one tile costs 1.20 BAM. A parent warns that cutting around a circle usually produces more waste than cutting along straight walls. The council must decide whether to approve the plan as proposed or to ask for a revised estimate. They also need a defensible number to present at the school board meeting, where any unexplained rounding will be questioned.
1. Calculate the net area to be tiled and the minimum number of tiles before waste. Show your rounding.
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Courtyard area = 12.6 × 8.4 = 105.84 m². Fountain area = π × 1.5² ≈ 7.07 m². Net area ≈ 105.84 - 7.07 = 98.77 m². One tile covers 0.30 × 0.30 = 0.09 m². Tiles before waste = 98.77 ÷ 0.09 ≈ 1097.4, so at least 1098 whole tiles.
2. Using the 8% allowance, determine the number of boxes, total tiles, and cost. Is the budget sufficient?
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8% allowance: 1098 × 1.08 = 1185.84, so 1186 tiles. Boxes: 1186 ÷ 25 = 47.44, so 48 boxes, which gives 1200 tiles. Cost = 1200 × 1.20 = 1,440 BAM. The budget of 1,500 BAM is sufficient, with 60 BAM left. Note that 48 boxes already provide 1200 - 1098 = 102 spare tiles, about 9.3% spare, slightly more than the 8% allowance.
3. The parent's warning leads the council to test a 12% allowance. Recalculate and advise the council. What should they do?
Read the worked answer
12% allowance: 1098 × 1.12 = 1229.76, so 1230 tiles. Boxes: 1230 ÷ 25 = 49.2, so 50 boxes, which gives 1250 tiles. Cost = 1250 × 1.20 = 1,500 BAM, exactly the budget. A 49-box purchase gives 1225 tiles, only 5 short of the 12% target but about 11.6% spare, and costs 1,470 BAM. Recommendation: approve 48 boxes only if the cutting plan is careful and the 8% figure is trusted; approve 49 boxes as a compromise if the council wants a near-12% buffer and a small cash reserve; approve 50 boxes only if the 12% buffer is non-negotiable, accepting zero contingency. The board should also ask whether breakage around the fountain can be reduced by obtaining a few loose tiles, since boxes are the only unit.