Painting a Hallway: Composite Area and Coverage
A facilities crew plans to repaint a rectangular hallway wall that is 36 feet long and 10 feet high. The wall contains two identical doors, each 3 feet wide and 7 feet high, and four identical windows, each 4 feet wide and 3 feet high. One gallon of paint covers 350 square feet per coat, and the crew will apply two coats. The crew plans to buy 10% extra paint for touch-ups and waste. Paint costs $28.50 per gallon and can be purchased only in whole gallons. A student volunteer calculates the paintable area by subtracting the perimeters of the doors and windows from the wall area, then divides by 350 and says one gallon is enough. The supervisor wants a correct calculation, a purchasing recommendation, and a short explanation of why the student's method confuses linear and square units.
1. Calculate the net paintable area of the wall after subtracting the doors and windows. Show each subtraction.
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The full wall area is 36 × 10 = 360 square feet.
Each door is 3 × 7 = 21 square feet. There are two doors, so the door area is 2 × 21 = 42 square feet.
Each window is 4 × 3 = 12 square feet. There are four windows, so the window area is 4 × 12 = 48 square feet.
The total opening area is 42 + 48 = 90 square feet. The net paintable area is 360 − 90 = 270 square feet.
The student's perimeter subtraction is not valid here. Perimeters measure length in feet, while the wall area is in square feet. Opening areas, not edge lengths, must be subtracted from the wall area.
2. Determine the number of gallons needed for two coats with 10% extra and the total cost. Explain the rounding.
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One coat needs 270 square feet of coverage. Two coats need 2 × 270 = 540 square feet.
Add 10% extra for touch-ups and waste: 10% of 540 is 54, so the required coverage is 540 + 54 = 594 square feet.
Each gallon covers 350 square feet per coat, so divide: 594 ÷ 350 ≈ 1.697. The crew cannot buy 0.697 of a gallon, and one gallon would cover only 350 square feet, which is less than 594. Therefore the crew must round up to 2 gallons.
The cost is 2 × $28.50 = $57.00. Rounding down would leave insufficient paint; rounding up to the next whole gallon is required by the coverage constraint.
3. The student says subtracting perimeters is acceptable because 'we are removing the edges where paint stops.' Write a response that identifies the unit error and describes a correct way to account for trim or edging if the crew also needs to paint a border around each opening.
Read the worked answer
Subtracting perimeters from an area mixes units. The wall area is in square feet, but a perimeter is in feet. A perimeter describes the distance around an opening, not the amount of wall surface removed by the opening. The correct subtraction is width × height for each rectangle.
If the crew also needs to paint a border around each opening, that border is a separate linear or area calculation, not part of the wall-area subtraction. The total edge length around the openings is: each door has perimeter 2(3 + 7) = 20 feet, so two doors give 40 feet; each window has perimeter 2(4 + 3) = 14 feet, so four windows give 56 feet. The total border length is 96 feet. If the border has a width, its area would be border width × border length. That trim calculation should be kept separate from the 270-square-foot wall area.