To understand how the metabolic rate of a 70.0 kg human compares to that of a 4.48 kg cat, we need to apply the given relationship that the metabolic rate is proportional to the mass raised to the power of [tex]\( \frac{3}{4} \)[/tex].
1. Determine the mass of the human and the cat:
- Human: [tex]\( 70.0 \)[/tex] kg
- Cat: [tex]\( 4.48 \)[/tex] kg
2. Calculate the metabolic rate for each:
- For the human:
[tex]\[
(70.0)^{\frac{3}{4}} \approx 24.2005
\][/tex]
- For the cat:
[tex]\[
(4.48)^{\frac{3}{4}} \approx 3.0793
\][/tex]
3. Find the factor by which the metabolic rate of the human exceeds that of the cat:
- Divide the metabolic rate of the human by the metabolic rate of the cat:
[tex]\[
\frac{24.2005}{3.0793} \approx 7.8590
\][/tex]
Thus, the metabolic rate of a 70.0 kg human exceeds that of a 4.48 kg cat by a factor of approximately [tex]\( 7.859 \)[/tex].
