Scaling Questions Key

Scaling Questions Key - 1)

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: 1) If
heart
rate
scales
with
a
‐0.25
exponent
and
stroke
volume
scales
with
and
 exponent
of
1.0
what
exponent
would
you
expect
for
cardiac
output?
 Cardiac
Output
is
proportional
to
body
mass
to
the
0.75
 
 
 
 
 
 
 2) What
scaling
exponent
would
you
expect
under
conditions
of
isometry
for
a
 plot
of
surface
are
per
unit
volume
(on
the
y‐axis)
and
volume
(on
the
x‐ axis)?

Explain
your
answer.
 The
scaling
exponent
is
‐1/3
 
 
 
 
 
 3) If
heart
rate
scales
proportional
to
body
mass
to
the
exponent
‐0.25
and
life
 span
scales
to
the
exponent
0.25,
write
the
scaling
relationship
for
the
 number
of
heart
beats
that
occur
during
an
organisms
life
time.

Explain
 algebraically
how
you
arrived
at
your
answer.
 The
scaling
relationship
for
the
number
of
heart
beats
to
body
mass
is
0.
 
 
 
 
 The
HB
does
not
change
with
body
mass
and
the
line
is
flat.
 
 4) Suppose
you
are
a
dendrologist
(one
who
studies
trees)
and
you
are
 measuring
the
relationship
between
wood
production
and
rainfall.

In
trees
 that
grow
in
wet
areas
the
measured
scaling
exponent
for
trunk
diameter
(d)
 versus
height
is
3.

In
trees
that
grow
in
dry
areas
the
measured
scaling
 exponent
for
trunk
diameter
versus
height
is
2.
 a) What
would
you
expect
the
scaling
exponent
for
the
mass
of
wood
 produced
(m)
versus
height
to
be
for
each
area?
 For
trees
in
wet
areas
 d = diameter h = height r = radius dαh 3 MassαVolume Volume = π × r 2 × h 
 r = 0.5 d rα 0.5 × h 3 Massα (0.5 × h 3 ) 2 × h Massα 0.25 h 6 × h Massα 0.25 h 7 For
trees
in
dry
areas
 € d = diameter h = height r = radius dαh 2 MassαVolume Volume = π × r 2 × h 
 r = 0.5 d rα 0.5 × h 2 Massα (0.5 × h 2 ) 2 × h Massα 0.25 h 4 × h Massα 0.25 h 5 
 € 
 b) If
trees
grown
in
dry
areas
have
enough
wood
for
harvest
at
30
years
of
 age,
at
what
age
can
trees
grown
in
wet
areas
be
harvested?
 In
order
to
answer
this
question
we
have
to
make
the
assumption
that
age
is
directly
 proportional
to
height.

If
height
is
proportional
to
age
then
we
can
substitute
age
in
 for
height
and
solve
for
the
equation.
 a = age h = height m = mass aαh mwet = mdry 5 7 5 7 0.25 hdry = 0.25 hwet 0.25 adry = 0.25 awet 0.25 × 30 5 = 0.25 awet 30 5 = awet 7 24, 300, 000 = awet 7 1 24, 300, 000 7 = awet 11.35 = awet € 
 7 
 5) You
are
an
elementary
school
teacher
and
you
find
that
your
student’s
test
 scores
follow
a
predictable
pattern.
 
 Score
=
(Rib
cage
volume
x
Height)/Surface
area
of
the
cranium
 
 
 How
would
you
expect
their
score
to
scale
with
body
mass
assuming
 isometry?
 m = mass v = volume h = height s = surfacearea vαm 1 hαm 3 2 
 sαm 3 1 Scoreα m × m3 2 m3 4 Scoreα m3 m Scoreαm 2 3 2 3 6) At
a
scientific
meeting,
a
presenter
showed
that
predatory
birds
have
larger
 eyes
(greater
diameter)
than
prey‐bird
species.

What
crucial
factor
should
 the
presenter
have
accounted
for
in
making
this
claim?

What
would
you
 expect
the
scaling
exponent
to
be
in
the
relationship
is
isometric?

The
 scaling
exponent
for
this
relationship
is
1
is
this
an
example
of
positive
or
 negative
allometry?
 € The
presenter
should
have
considered
that
predatory
birds
tend
to
belarger
than
 prey
species.

It
would
be
expected
that
larger
birds
would
have
larger
eyes.


 Diameter
is
a
length
therefore
 1 3 DiameterαMass 
 Since
the
slope
of
the
relationship
is
greater
than
1/3
this
is
an
example
of
positive
 allometry.
 € 
 
 
 7) Peak
stress
in
the
leg
bones
during
locomotion
is
predicted
to
increase
 proportional
to
body
mass
for
quadrupedal
(i.e.
four‐footed)
mammals.
 Below
is
a
graph
from
a
paper
on
safety
factors
and
body
size
(Biewener,
 1982).
 
 a) 
 a)
Estimate
the
equation
of
the
regression
line.
 Slope
approximately
0.75
 b) What
exponent
for
the
allometric
equation
would
you
expect
for
an
isometric
 relation
between
area
an
body
mass?
 Scales
to
the
2/3
 c) Does
the
graph
indicate
positive
or
negative
allometry
relative
to
your
 isometrice
prediction
from
b)?
 Positive
allometry
 d) Does
your
answer
from
c)
imply
that
larger
animals
will
have
relatively
more
 slender,
stout
or
isometrically
similarly
shaped
leg
bones?

 Stouter
Legs
 e) What
bone
diameter
would
you
predict
for
a
formidable
rat,
with
a
body
 mass
of
1kg?
 3.56
mm
 
 ...
View Full Document

This note was uploaded on 03/29/2012 for the course LS 2 taught by Professor Andrewa.biewener,petert.ellison,anddaniele.lieberman during the Fall '10 term at Harvard.

Ask a homework question - tutors are online