The production of heat by metabolic processes takes place throughout the volume

Question

The production of heat by metabolic processes takes place throughout the volume of an animal, but loss of heat takes place only at the surface (i.e. the skin). Since heat loss must be balanced by heat production if an animal is to maintain a constant internal temperature, the relationship between surface area and volume is relevant for physiology. If the surface area of a cube is increased by a factor of 2 (so that the new surface area is twice the size of the original surface area), by what factor does the volume of the cube change? Give your answer to two significant figures. 1.59

Explanation / Answer

we know that surface area of the cube is A = 6*a^2

where a is the length of the each side of the cube

given that new surface area is A2 = 2*A1 = 2*6*a1^2 = 12*a1^2

6*a2^2 = 12*a1^2

a1 is the original length of each side of the cube

and a2 is the new length of each side of the cube

a2 = sqrt(2)*a1

original volume of cube is a1^3

new volume of the cube is a2^3 = (sqrt(2)*a1)^3 = 2.83*a1^3

change in volume of the cube is (2.83-1)*a1^3 = 1.83*a1^3

so required factor is 1.83

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