64
4
4.1 (a) Since 2.54 cm
5
1 inch, inches are changed to centimeters by multiplying by 2.54. Letting
x
5
height in inches and
y
5
height in centimeters, the transformation is
y
5
2.54
x,
which is
monotonic increasing. (b) If
x
5
typing speed in words per minute, then
5
typing speed in
words per second and
number of seconds needed to type a word. The transformation is
, which is monotonic decreasing (we can ignore the case
x
,
0 here). (c) Letting
x
5
diame-
ter and
y
5
circumference, the transformation is
y
5
p
x,
which is monotonic increasing. (d)
Letting
x
5
the time required to play the piece, the transformation is
y
5
(
x
2
5)
2
, which is not
monotonic.
4.2 Monotonic increasing: intervals 0 to
and
to 2
p
. Monotonic decreasing: interval
to
.
4.3 Weight
5
c
1
(height)
3
and strength
5
c
2
(height)
2
; therefore, strength
5
c
(weight)
2/3
, where
c
is a constant.
4.4 A graph of the power law
y
5
x
2/3
(see below) shows that strength does
not
increase linearly
with body weight, as would be the case if a person 1 million times as heavy as an ant
could
lift 1
million times more than the ant. Rather, strength increases more slowly. For example, if weight is
multiplied by 1000, strength will increase by a factor of (1000)
2/3
5
100.
3
p
2
p
2
3
p
2
p
2
y
5
60
x
y
5
60
x
5
1
60
x
4.5 Let
y
5
average heart rate and
x
5
body weight. Kleiber’s law says that total energy con-
sumed is proportional to the three-fourths power of body weight, that is, Energy
5
c
1
x
3/4
. But
total energy consumed is also proportional to the product of the volume of blood pumped by the
heart and the heart rate, that is, Energy
5
c
2
(volume)
y
. Furthermore, the volume of blood
pumped by the heart is proportional to body weight, that is, Volume
5
c
3
x
. Putting these three
equations together yields
c
1
x
3/4
5
c
2
(volume)
y
5
c
2
(
c
3
x
)
y
. Solving for
y,
we obtain
.
y
5
c
1
x
3
>
4
c
2
c
3
x
5
cx
2
1
>
4
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