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Thank you for looking at my question.

We're supposed to paint the walls of a room and ceiling twice (x2), the room is 15 ft long by 10 ft wide and 8 ft tall. The teacher is asking us to provide the total area in square feet.

The answer given is 550 sq ft (i imagine that includes the ceiling?)

To me area is length times width (15 * 10) * 2 (in this case). I'm not sure what to do with that 8. 15*10*8=1200 ft cubed? How does the fact that I have to paint the room twice factor into math??

How did the teacher arrive to that result?

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$15\times 10+2(15\times 8)+2(10\times 8)$. –  J. M. Oct 21 '11 at 12:29
    
Thank you very much. I was on the wrong track. –  anon_ying Oct 21 '11 at 16:11
    
Would you mind expanding on how you got to that answer? –  anon_ying Oct 21 '11 at 16:20
    
joriki's answer more or less hints at how you should be assembling the expression that gives the area you need. Schematically, I gave you $\text{ceiling}+2\text{ walls}+2\text{ walls}$... –  J. M. Oct 21 '11 at 16:31

2 Answers 2

You didn't specify what you're being asked to calculate the "total area" of. If it's the total area of the walls and the ceiling, then obviously the fact that you want to paint them twice doesn't change that area. On the other hand, the question might be about the total area of paint covers to be applied. That would make sense for instance if you want to figure out how much paint to buy, since the amount of paint you need is proportional to the total area of paint covers you want to apply. If you want to paint surfaces with a certain total area twice, then the area of the paint covers to be applied is just twice the total area of the surfaces to be painted. However, from the answer you give, it seems that you're being asked to calculate the total area of the walls and the ceiling; in that case it's irrelevant how often you intend to paint them.

Regarding your attempt at calculating the surface area: You need to think more about the meaning of the terms "length" and "width". You're mixing up two different contexts in which these occur. When you calculate the area of a rectangle as "length times width", these terms refer to the two dimensions of the rectangle, independent of its position or orientation in space. By contrast, when we talk about the dimensions of a room, we speak of the room's length, width and height, meaning, respectively, its longer horizontal dimension, its shorter horizontal dimension and its vertical dimension. You equated these different meanings of the terms and just multiplied the length and width of the room.

What you actually need to do is to add up the areas of all the rectangles forming the four walls and one ceiling of the room. These will all be the product of the length and width of those individual rectangles, but these dimensions will be different combinations of the length, width and height of the room. Just go through all the walls and the ceiling and figure out which of the three given dimensions form the two dimensions of each of the rectangles.

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This was actually really helpful. I was on the wrong track. This comment helped me realize that I had to treat the wall and the celing differently. Thank you. –  anon_ying Oct 21 '11 at 16:10

Well the volume of the room won't be too useful, unless you paint the room by filling it with paint, which might work but isn't what is being asked here.

Think of this as a surface area problem - instead of finding "the area" to be 10*15, think of how you would paint the room - each wall, then the ceiling, then each wall again, then the ceiling again. So the area of each of the four walls and the ceiling would need to be painted, each twice.

I'm not sure about that answer though, it looks like that is number of square feet required to paint the room once, not twice.

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