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A study finds that the metabolic rate of mammals is proportional to [tex]$m^{3 / 4}$[/tex], where [tex]$m$[/tex] is the total body mass. By what factor does the metabolic rate of a [tex]$70.0 \, \text{kg}$[/tex] human exceed that of a [tex][tex]$4.48 \, \text{kg}$[/tex][/tex] cat?

Sagot :

GinnyAnswer
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].
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