The blue whale ( Balaenoptera musculus) is the largest mammal and it has a mass of 136,000 kg. The smallest mammal is Kitti's hog-nosed bat (Craseonycteris thonglongyai) with a mass of only 2 g. Assuming both animals obey the above scaling relationship, what is the ratio of the heartbeat frequencies of Kitti's hog-nosed bat to the blue whale? Report your answer uisng two significant figures.

To solve this problem, we'll use the allometric scaling relationship for heart rate in mammals, which is generally given by:

\[
f \propto M^{-\frac{1}{4}}
\]

Here, \( f \) is the heartbeat frequency, and \( M \) is the mass of the animal. The ratio of the heartbeat frequencies for two animals with masses \( M_1 \) and \( M_2 \) can be calculated as:

\[
\frac{f_1}{f_2} = \left(\frac{M_2}{M_1}\right)^{\frac{1}{4}}
\]

Given:
- Mass of the blue whale (\( M_1 \)) = 136,000 kg = 136,000,000 g
- Mass of Kitti's hog-nosed bat (\( M_2 \)) = 2 g

The ratio of the masses is:

\[
\frac{M_2}{M_1} = \frac{2 \text{ g}}{136,000,000 \text{ g}} = \frac{2}{136,000,000}
\]

The ratio of the heartbeat frequencies is:

\[
\frac{f_2}{f_1} = \left(\frac{2}{136,000,000}\right)^{\frac{1}{4}}
\]

Now, let's calculate this ratio:

\[
\frac{f_2}{f_1} = \left(\frac{2}{136,000,000}\right)^{\frac{1}{4}} \approx \left(\frac{1}{68,000,000}\right)^{\frac{1}{4}} \approx \left(1.47 \times 10^{-8}\right)^{\frac{1}{4}} \approx 0.0107
\]

Finally, take the reciprocal to get the ratio of the heartbeat frequencies of Kitti's hog-nosed bat to the blue whale:

\[
\frac{f_{\text{bat}}}{f_{\text{whale}}} \approx \frac{1}{0.0107} \approx 93.5
\]

To report the answer with two significant figures:

\[
\boxed{94}
\]

Thus, the ratio of the heartbeat frequencies of Kitti's hog-nosed bat to the blue whale is approximately 94.

Would you like further details or have any questions about the calculations? Here are five related questions you might find interesting:

1. How does the mass of an animal affect its metabolic rate?
2. Why do smaller animals generally have faster heart rates than larger animals?
3. Can the scaling relationship \( f \propto M^{-\frac{1}{4}} \) be applied to non-mammals?
4. How does body size impact the lifespan of different species?
5. What other physiological processes in animals are affected by scaling laws?

**Tip:** Understanding scaling relationships can provide insights into how different physiological processes are interconnected in animals of varying sizes.

Learn how to calculate the heartbeat frequency ratio between Kitti's hog-nosed bat and the blue whale using allometric scaling. Understand the scaling relationship and its application in mammalian physiology. Suitable for advanced high school students studying biology or mathematics.

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