Thermodynamics: Heating a Rectangular Box

[spockjones20]

Homework Statement

This question is from a thermodynamics test from a previous science olympiad competition that I am using to study from for a future test.
"Consider two neighboring rectangular houses built from the same materials. One of the houses has twice the length, width, and height of the other. Under identical climatic conditions, what would be true about the rate that heat would have to be supplied to maintain the same inside temperature on a cold day? Compared to the small house, the larger house would need heat supplied at:
A.) twice the rate
B.) 4 times the rate
C.) 16 times the rate
D.) 8 times the rate

Homework Equations

Volume of small box = l*w*h
Volume of large box = 2l*2w*2h = 8(l*w*h)
I honestly do not know if any other equations need apply here, being new to thermo I thought that it would just be a ratio of the volumes

The Attempt at a Solution

I thought that the answer should be D, 8 times the rate because the volume of the larger box is 8 times that of the smaller box. However, the test has the answer of B, four times the rate. I am not sure if this is a mistake in the key or if there is another equation that I should be using. Help is very much appreciated.

 

[haruspex]

How does heat escape a house? What route does it take?

 

[spockjones20]

Well I am assuming it does not go through the floor, so that might get taken out. Heat rises, so would I focus on only the height aspect?

 

[fortissimo]

The heat radiated is proportional to the surface area of the walls, so the answer is B.

 

[haruspex]

The heat radiated is proportional to the surface area of the walls, so the answer is B.

Quite so, but the preferred style on these forums is to guide people into figuring things out for themselves, not just providing the answer.

 

[spockjones20]

Ok this makes sense. So if they asked about heating a room, instead of maintaining a heat, would the volume play a bigger role? I can see how the loss refers to the surface area now, since it would not be lost from the inside of the box (the volume). Thank you

 

[haruspex]

Ok this makes sense. So if they asked about heating a room, instead of maintaining a heat, would the volume play a bigger role?

If you mean heating it up from a lower temperature, yes, but even then perhaps not such a great role. Warming a room involves heating the air, but also the walls etc.