Allometric functions of the form
are often used by experimentalists in biology and ecology. This exercise explores such an application, and the data are given in Table 8.9. What is given, for each animal, is its typical mass and its maximum relative speed. The latter is the animal’s maximum speed divided by its body length. This exercise uses the results from Exercise 8.6(a),(b).
(a) Taking
to be the mass, fit the power law to the data in Table 8.9 and then plot the data and power law curve using a log-log plot.
(b) Based on your result from part (a), what was the running speed of a Tyrannosaurus rex?
(c) Taking
to be the relative speed, fit the power law to the data in Table 8.9 and then plot the data and power law curve using a log-log plot.
(d) If speed =
(mass)β, then mass =
(speed)
b
, where
= 1/
. Based on this, one might think that the exponents from parts (a) and (c) satisfy
= 1/. Do they? Using a sketch similar to the ones in Figure 8.5, but in the
-plane, explain why the error functions are different in the two cases. Because of this, it is not expected that
= 1/.
Exercise 8.6(a),(b)
(a) Writing
=
1
v2
, and then taking the log of this equation, show that the transformed model function can be written as
(
) =
1
+
2
, where
1
= log
1
and
2
=
2. Also, show that the transformed data points (
i
,
i) are
i
= log
i
and
i
= log
i
.
(b) Continuing from part (a), using the least squares error
(
1,
2) =
(
1
+
2
i
−
i
)2, and the common log, show that
1
= 10
V1
and
2
=
2, where
1
and
2
are given in (8.31).