Body Weight and Height, Strength and Agility

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Body Weight and Height, Strength and Agility

• Section 2.5

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Body Weight and Height

• How much should I weigh?
• A rule of thumb for marathon runners is 2 lb of
  body weight per inch of height.
• Organizations, such as the Army, are concerned
  about physical conditioning and define an upper
  weight allowance acceptability.

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Body Weight and Height

• No other delineators such as bone density.
• Note in a large portion of the table 5 lb per
  inch is allowed.

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Body Weight and Height

• In this section we examine qualitatively how
  weight and height should vary.
• Body weight depends on a number of factors
  Height is one factor.
• Bone density could be another. Significant
  variation or essentially constant? Bone volume?
  
• Body density factor. Differences in densities of
  bone, muscle, and fat. Is body density a
  function of age and gender?

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Body Weight and HeightProblem Identification

• Define the problem so that bone density is
  constant (by accepting an upper limit) and
  predict weight as a function of height, gender ,
  age and body density.
• The purpose of the weight table must also be
  specified, so we will base the table on physical
  appearance.

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Body Weight and HeightProblem Identification

• We identify the problem as follows
• For various heights, genders, and age groups,
  determine upper weight limits that represent
  maximum levels of acceptability based on physical
  appearance.

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Body Weight and HeightAssumptions

• Assumptions about body density
• One simplifying assumption Assume that some
  parts of the body are composed of an inner core
  of a different density.
• Assume the inner core is composed primarily of
  bones and muscle and the outer core is primarily
  a fatty material of a different density.

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Body Weight and HeightAssumptions

• Assume that for adults certain parts of the body,
  such as the head, have the same volume and
  density for different people.
• So the weight of an adult is given by
• W k1 Win Wout, where k1 is the constant
  weight of those parts having the same volume and
  density for different individuals. Win and Wout
  are the weights of the inner and outer cores,
  respectively.

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Body Weight and HeightAssumptions Submodel
inner core

• We will now consider a submodel for the inner
  core.
• People are not geometrically similar.
• However, we are concerned with an upper weight
  limit based on physical appearance. So it would
  seem reasonable that whatever image might be
  visualized as an upper limit standard of
  acceptability for a 74-in person would be a
  scaled image of a 65-in person.
• Thus for our problem, geometrical similarity of
  individuals is a reasonable assumption.

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Body Weight and HeightAssumptions Submodel
inner core

• So Vin ? h3
• Now, what should be the average weight density of
  the inner core?
• Assuming the inner core is composed of muscle and
  bone, each of different weight densities, what
  percentage of the total volume of the inner core
  is occupied by the bones?

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Body Weight and HeightAssumptions Submodel
inner core

• If the bone diameter is assumed to be
  proportional to the height, then the total volume
  occupied by the bones is proportional to the cube
  of the height. This implies that the percentage
  of the total volume of the inner core occupied by
  bones in geometrically similar individuals is
  constant.
• And it follows that weight density ?in is
  constant (as shown in the following slide).

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Body Weight and HeightAssumptions Submodel
inner core

• Consider the average weight density ?avg of a
  volume V consisting of two components V1 and V2,
  each with a density ?1 and ?2.
• Then V V1 V2 and
• ?avgV W ?1V1 ?2V2, thus
• ?avg ?1(V1/ V) ?2(V2/ V).
• Thus ?avg is constant, so long as V1/ V and V2 /
  V do not change.

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Body Weight and HeightAssumptions Submodel
inner core

• Application of this result to the inner core
  implies that the average weight density ?in is
  constant.
• Thus Win Vin ?in ? h3, or
• Win k2 h3, for k2 gt 0
• Note that the preceding submodel includes any
  case of materials with densities different than
  muscles and bone (such as tendons, ligaments, and
  organs) as long as their percentage of the total
  volume of the inner core is constant.

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Body Weight and HeightAssumptions Submodel
outer core

• Now consider the outer core of fatty material.
• It can be argued that the thickness of the outer
  core should be constant regardless of the height.
  
• Let ? denote the thickness.
• Then Wout ??outSout, where Sout is the surface
  area of the outer core and ?out is the density of
  the outer core.

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Body Weight and HeightAssumptions Submodel
outer core

• Repeating Wout ??outSout
• Again assuming the subjects are geometrically
  similar Sout ? h2
• So Wout ? h2.
• It may be argued, however, that taller people can
  carry a greater thickness for the fatty layer.
  If it is assumed that the thickness of the outer
  core is proportional to the height, then Wout ?
  h3.

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Body Weight and HeightAssumptions Submodel
outer core

• Allowing both these assumptions to reside in a
  single submodel gives Wout k3h2 k4h3, where
  k3, k4 ? 0.
• Summing the two submodels Win k2 h3, for k2 gt
  0 andWout k3h2 k4h3, where k3, k4 ? 0,in W
  k1 Win Wout .
• We get W k1 k3h2 k5h3, for k1, k5 gt 0 and
  k3 ? 0, where k5 k2 k4.

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Body Weight and HeightModel

• Model W k1 k3h2 k5h3.
• Note that the model suggests variations in weight
  of a higher order than the first power of h. If
  the model is valid, then taller people will
  indeed have a difficult time satisfying the
  linear rules given earlier.
• Our judgments can only be qualitative since we
  have not verified the submodels.

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Body Weight and HeightModel Interpretation

• Lets interpret the general rules given earlier,
  which allowed a constant weight increase for each
  additional inch of height, in terms of our
  submodel.
• Because the total allowable weight increase per
  inch is assumed constant by the given rules, the
  portion allowed for the trunk increase may also
  be assumed constant.

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Body Weight and HeightModel Interpretation

• To allow a constant weight increase, the trunk
  must increase in length while maintaining the
  same cross-sectional area. This implies, for
  example, that the waist size remains constant.
• If a 30-in. waist is judged the upper limit
  acceptable for the sake of personal appearance in
  a male with a height of 66 in., then the 2 lb per
  inch rule would allow a 30-in. waist for a male
  with a height of 72 in. as well.

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Body Weight and HeightModel Interpretation

• On the other hand, the model based on geometric
  similarity suggests that all distances between
  corresponding points should increase by the same
  ratio. Thus, the male with a height of 72 in.
  should have a waist of 30(72/66) ? 32.7 in.

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Body Weight and HeightModel Interpretation

• Now we can see why tall marathoners who follow
  the 2 lb per inch rule appear very thin.

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Strength and Agility

• Consider a competitive sports contest in which
  men or women of various sizes compete in events
  emphasizing strength (such as weight lifting) or
  agility (such as an obstacle course).
• How would you handicap such events?

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Strength and AgilityProblem Identification

• For various heights, weights, genders, and age
  groups, determine their relationship with agility
  in competitive sports.

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Strength and AgilityAssumptions

• Lets initially neglect gender and age.
• We assume agility is proportional to the ratio
  strength / weight.
• We further assume that strength is proportional
  to the size of the muscles being used in the
  event, and we measure this size in terms of the
  muscles cross-sectional area.

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Strength and AgilityAssumptions

• Recall that weight is proportional to volume
  (assuming constant weight density).
• If we assume all participants are geometrically
  similar, we have
• Agility ? strength / weight ? l2 / l3 ? 1 / l

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Strength and AgilityAssumptions

• Agility ? strength / weight ? l2 / l3 ? 1 / l
• This is a nonlinear relationship between agility
  and the characteristic dimension, l.
• We also see that under these assumptions agility
  has a nonlinear relationship with weight.
• How would you collect data for this model?
• How would you test and verify your model?