Staining a Deck with a Semicircular Cutout
A homeowner is staining a rectangular redwood deck measuring 18 ft by 24 ft. A semicircular hot tub cutout with a 10-ft diameter is removed from one of the 24-ft sides, so the stained area is the rectangle minus the semicircle. Stain is sold in 1-gallon cans that cover 350 ft² per coat and 1-quart cans that cover 100 ft² per coat. A gallon costs $38 and a quart costs $12, and only whole containers can be purchased. The manufacturer requires two coats. The homeowner's brother estimates that 'about two gallons' will be enough and plans to buy two gallon cans. The homeowner also needs rubber edging for the curved part of the cutout; edging is sold in 8-ft strips. Both the stain and the edging are in stock.
1. Compute the area of the deck that will be stained. Then determine the total square footage of stain needed for two coats.
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Rectangle area = 18 × 24 = 432 ft². The cutout is a semicircle of diameter 10 ft, so radius = 5 ft, and its area is (1/2)π(5²) = 12.5π ≈ 39.27 ft². Deck area = 432 − 39.27 ≈ 392.73 ft². Two coats require 2 × 392.73 ≈ 785.46 ft² of coverage. Using π ≈ 3.14 gives 12.5π ≈ 39.25 and 785.5 ft²; the small difference does not change the container decision.
2. The brother plans to buy two 1-gallon cans. Determine whether that is enough, and find the least expensive combination of gallon and quart cans that covers two coats.
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Two gallons cover 2 × 350 = 700 ft², which is less than 785.46 ft², so two gallons are not enough. Test combinations: 3 gallons cover 1050 ft² and cost 3 × 38 = $114. 2 gallons + 1 quart cover 700 + 100 = 800 ft² and cost 76 + 12 = $88. 2 gallons + 2 quarts cost $100 and cover 900 ft². 1 gallon + 5 quarts cost 38 + 60 = $98 and cover 850 ft². 8 quarts cost $96 and cover 800 ft². The minimum is $88 for 2 gallons and 1 quart, which provides 800 ft², enough for 785.46 ft² with a small margin. This shows why you compare total two-coat area and round up to whole containers.
3. If the deck absorbs 10% more stain than the manufacturer's coverage figure, how much stain is needed, and what is the least expensive purchase? Also determine how many 8-ft strips of edging are needed for the curved part of the cutout.
Read the worked answer
With 10% waste, required coverage = 785.46 × 1.10 ≈ 864.0 ft². Now 2 gallons + 1 quart covers 800 ft², which is short. 2 gallons + 2 quarts covers 900 ft² and costs 76 + 24 = $100. 3 gallons covers 1050 ft² and costs $114. 8 quarts covers 800 ft² and is short; 9 quarts covers 900 ft² and costs $108. So the least expensive is 2 gallons + 2 quarts at $100. For the edging, the curved edge is a semicircle of diameter 10 ft: arc length = (1/2)π(10) = 5π ≈ 15.71 ft. Each strip is 8 ft, so 1 strip is too short, and 2 strips provide 16 ft, enough. The homeowner needs 2 strips.