Mass specific metabolic rate (R/M) is then:
These two relationships are straight lines if we plot log(R) or log(R/M) vs. log(M). (Eckert, Fig. 16-7c) The slopes of these straight lines are b and (b-1), respectively. IMPORTANT - Equation 16-6 on pg. 677 is wrong!
If we are comparing two spherical bodies, l is the radius and surface area = 4 Pi l2 (c = 4 Pi). If we double the radius, the surface area increases by 4 times. If we are comparing two cubes, l is the width and height of each one of the faces of the cube and surface area = 6 l2 (c = 6). If we double the linear dimension of each face, the surface area of each face, and thus of the cube, increases by 4 times.
Volume (V) is proportional to linear dimension (l) in the following way:
If we are comparing two spherical bodies, volume = (4/3) Pi l3 (d = (4/3) Pi). If we double the radius, the volume increases by 8 times. If we are comparing two cubes, volume = l3 (d = 1). If we double the width of one face, the volume increases by 8 times. Mass is also proportional to the cube of linear dimension since if we have more volume and the density of material remains the same, so, M = e l3, where e is just another proportionality constant.
Now, let's take the log of the surface and the mass relationships:
So, if we plot log of the animal's surface area vs. log of the animal's mass for different animals of different sizes (different l's), as we increase l (the animal becomes longer, higher, and deeper in the same proportion), the log(S) increases by 2 units while at the same time log(M) increases by 3 units. In other words, the slope of the log(S) vs. log(M) relationship is 2/3 or 0.67. This is for an isometric relationship. Only adults of the same species obey this relationship.
When we compare many different species, the log(S) vs. log(M) relationship has a slope of 0.63 instead of 0.67, because of the thickening of bones and other body structures in larger animals (allometric scaling - proportions change as overall size changes). (Eckert, Fig. 16-9)
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