Brick Border Around a Patio
A homeowner has a rectangular concrete patio measuring 16 ft by 10 ft and wants to surround it on all four sides with a border of uniform width made of brick pavers. The supplier has delivered 120 square feet of pavers, and the homeowner wants to use essentially all of them on the border, so the border must have the same width on every side. The pavers are sold in boxes covering 12.5 square feet each, and the homeowner intends to buy 10% extra to allow for cuts and breakage. Edging strip for the outer perimeter of the finished patio-plus-border is sold by the linear foot.
1. Find the width of the border that uses exactly 120 square feet of pavers.
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Let x be the border width in feet. The finished outer rectangle is $(16 + 2x)$ by $(10 + 2x)$.
$$(16+2x)(10+2x) - 160 = 120$$ $$160 + 32x + 20x + 4x^2 - 160 = 120$$ $$4x^2 + 52x - 120 = 0$$ $$x^2 + 13x - 30 = 0$$ $$(x + 15)(x - 2) = 0$$
$x = 2$ or $x = -15$. A negative width is impossible here, so x = 2 ft.
Check: outer dimensions 20 ft by 14 ft = 280 sq ft; $280 - 160 = 120$ sq ft. The border is $2(20 \times 2) + 2(10 \times 2) = 80 + 40 = 120$ sq ft, matching the strip-by-strip count.
2. How many linear feet of edging are needed for the outside of the finished patio-plus-border?
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Outer dimensions from the previous part are 20 ft by 14 ft.
$$P = 2(20) + 2(14) = 40 + 28 = 68 \text{ ft}$$
68 linear feet. Do not use the original 16-by-10 patio (52 ft) - the edging wraps the outer edge of the brick, not the concrete.
3. How many boxes of pavers should be purchased, and what is the consequence of rounding at the wrong step?
Read the worked answer
Add the waste allowance before converting to boxes:
$$120 \times 1.10 = 132 \text{ sq ft}$$ $$132 \div 12.5 = 10.56 \text{ boxes}$$
Round up to whole boxes: 11 boxes (137.5 sq ft).
Rounding at the wrong step is the trap: $120 \div 12.5 = 9.6$, which rounds to 10 boxes, and 10 boxes supply only 125 sq ft - less than the 132 sq ft needed after waste. Ten boxes would leave the border short even before any breakage, which matters because the uniform-width design cannot tolerate a partially finished side.