Calculus/Advanced Math Allometry Problem Scaling and Allometry practice problems

Question

Calculus/Advanced Math Allometry Problem

Scaling and Allometry practice problems (1) Write the power law you would expect between each of the following quantities. Leave k as an unknown constant but estimate a value for p. There are two possible answers in each case (a) length and area (b) area and volume (c) volume and length (d) mass and area (e) energy exchange and mass (f) maximum weight supported and length (g) maximum weight supported and mass (2) If we double the volume, what happens to the mass? (3) If we double the volume, what happens to the area? (4) If we triple the mass, what happens to the energy exchange? (5) If we half the radius, what happens to the maximum weight supported? (6) Make a graph using the following table to estimate the power law relationship between the surface temperature of a mammal to its total mass surface temp (QC) body mass (kg) 25 62.5 27 70 30 82 35 103.5 37 112.5

Explanation / Answer

1)

a)

example circle area = pi*r2  

thus area = K length 2

b)

both area and volume is related by length

Volume= k (length)3 = K (length 2 ) 1.5 = k ( Area )1.5

c) volume of cube of side a ia a3

thus voume= K(length)3

d)

Areal density = mass /area

if areal density is constant of material then

Mass = K (Area)2

e)

Einstien relation E=mc2

Energy=K ( mass)

2)

density = mass/volume

volume = K mass

if we double the volume mass will be double

3)

volume = k ( area ) ^(1.5 )

area =K volume^(-1.5)

if we double the volume area= (2)^(-1.5)

4)

E=K*mass

if we tripple the mass the energy exchange will be triple

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