Paint Coverage, Cost, and Test Data
Rosa is painting a rectangular room that is 16 feet long, 20 feet wide, and 9 feet high. She will paint the four walls and the ceiling, but not the floor. The room has two windows, each 3 feet by 5 feet, and one door 3 feet by 7 feet; these openings are not painted. She plans to apply two coats everywhere. A gallon of paint is labeled to cover 350 square feet, but Rosa's test patches on similar walls covered 330, 345, 310, and 335 square feet per gallon. Paint costs $32 per gallon, and a quart costs $9 and covers 87.5 square feet. Sales tax is 7%.
1. Calculate the paintable area, including both coats. Using the label coverage of 350 square feet per gallon, how many gallons are needed if Rosa buys only whole gallons? Include tax in the cost.
Read the worked answer
Wall area: $2(16 + 20)(9) = 2(36)(9) = 648$ square feet.
Ceiling area: $16(20) = 320$ square feet.
Total before openings: $648 + 320 = 968$ square feet.
Openings: two windows are $2(3)(5) = 30$ square feet; the door is $3(7) = 21$ square feet; total $51$ square feet.
Paintable area: $968 - 51 = 917$ square feet. Two coats require $2(917) = 1834$ square feet.
At 350 square feet per gallon: $1834 \div 350 \approx 5.24$, so she must buy 6 whole gallons.
Cost: $6(32) = 192$ dollars. Tax is $0.07(192) = 13.44$, so the total is $205.44$.
2. Compare buying 6 whole gallons with buying 5 gallons plus 1 quart. Which is cheaper, and does the smaller purchase cover the job? Include tax.
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Five gallons plus one quart cover
$5(350) + 87.5 = 1750 + 87.5 = 1837.5$ square feet.
That is just enough for the 1834 square feet needed, with only 3.5 square feet to spare.
Cost: $5(32) + 9 = 160 + 9 = 169$ dollars. Tax is $0.07(169) = 11.83$, so the total is $180.83$.
The 6-gallon purchase costs $205.44, so 5 gallons plus 1 quart saves $205.44 - 180.83 = 24.61$ and technically covers the job, though with almost no margin.
3. Using the mean of Rosa's test-patch data, recompute the coverage needed. Does the 5-gallon-plus-quart purchase still cover the job comfortably? What would you recommend?
Read the worked answer
Mean test coverage:
$\frac{330 + 345 + 310 + 335}{4} = \frac{1320}{4} = 330$ square feet per gallon.
At that rate, the paint needed is
$1834 \div 330 \approx 5.56$ gallons.
The 5-gallon-plus-quart purchase is equivalent to 5.25 gallons. At 330 square feet per gallon, it covers
$5.25(330) = 1732.5$ square feet,
which is about 101.5 square feet short. It would not cover the job comfortably; it would likely run out during the second coat.
Recommendation: buy 6 gallons, which covers $6(330) = 1980$ square feet and leaves a reasonable margin. Alternatively, 5 gallons plus 3 quarts gives 5.75 gallons, covering $5.75(330) = 1897.5$ square feet, but 6 gallons is the simpler and safer choice.