Problem 94
Question
ALLOMETRY: Heart Rate It is well known that the hearts of smaller animals beat faster than the hearts of larger animals. The actual relationship is approximately (Heart rate) \(=250(\text { Weight })^{-1 / 4}\) where the heart rate is in beats per minute and the weight is in pounds. Use this relationship to estimate the heart rate of: A 625 -pound grizzly bear.
Step-by-Step Solution
Verified Answer
The estimated heart rate of a 625-pound grizzly bear is 50 beats per minute.
1Step 1: Understand the formula
The given formula for heart rate as a function of weight is \( \text{Heart rate} = 250 \times \text{Weight}^{-1/4} \). This formula tells us that heart rate is proportional to the weight raised to the power of \(-1/4\). We will use this formula to estimate the heart rate for a 625-pound grizzly bear.
2Step 2: Identify the known value
In this problem, the known value is the weight of the grizzly bear, which is given as 625 pounds.
3Step 3: Substitute the known weight into the formula
Replace the 'Weight' variable in the formula with 625. The formula becomes: \( \text{Heart rate} = 250 \times 625^{-1/4} \).
4Step 4: Compute the power of negative quarter
Calculate 625 raised to the power of \(-1/4\). This means you need to compute the fourth root of 625 and then find its reciprocal. The fourth root of 625 is 5, as \( 5^4 = 625 \). Hence, \( 625^{-1/4} = \frac{1}{5} \).
5Step 5: Calculate the heart rate
Now that we have \( 625^{-1/4} = \frac{1}{5} \), substitute back into the formula: \( \text{Heart rate} = 250 \times \frac{1}{5} = 50 \).
Key Concepts
Heart RateMathematical ModelingExponential FunctionsBiological Scaling Laws
Heart Rate
Heart rate is a measure of how many times the heart beats in one minute. It is a crucial indicator of cardiovascular health and varies among different species. Typically, smaller animals have hearts that beat faster than those of larger animals. For example, a mouse has a very high heart rate compared to a grizzly bear. This difference is primarily due to the varying metabolic needs of animals of different sizes, with smaller animals requiring more energy per unit of body mass.
In mathematical terms, heart rate can be depicted using various functions that describe its relationship with the animal's body weight. Understanding this relationship aids in the study of animal physiology and can even have applications in comparative biology and medicine.
In mathematical terms, heart rate can be depicted using various functions that describe its relationship with the animal's body weight. Understanding this relationship aids in the study of animal physiology and can even have applications in comparative biology and medicine.
Mathematical Modeling
Mathematical modeling involves creating equations and formulas that represent real-world phenomena. In the study of biology, these models help in understanding complex processes and structures by simplifying them into mathematical expressions.
The heart rate formula in the problem, which is \(\text{Heart rate} = 250 \times \text{Weight}^{-1/4}\), is a perfect example of this. It simplifies the relationship between heart rate and body weight into an easily interpretable mathematical connection. This model aids researchers in predicting heart rates across various species, facilitating a deeper understanding of biological patterns.
Such models are essential in biology and other sciences because they:
The heart rate formula in the problem, which is \(\text{Heart rate} = 250 \times \text{Weight}^{-1/4}\), is a perfect example of this. It simplifies the relationship between heart rate and body weight into an easily interpretable mathematical connection. This model aids researchers in predicting heart rates across various species, facilitating a deeper understanding of biological patterns.
Such models are essential in biology and other sciences because they:
- Provide clear insights into patterns and trends.
- Help in making predictions based on existing data.
- Simplify complex relationships into manageable equations.
Exponential Functions
Exponential functions are mathematical expressions where a variable, such as weight in our context, is raised to a specific power. These functions are instrumental in describing processes that exhibit rapid change or growth.
In the allometry problem, the use of the exponent \(-1/4\) indicates an inverse relationship between heart rate and weight. As weight increases, the exponent causes the heart rate to decrease. This is a typical behavior seen when modeling natural processes where rapid changes occur. For example, exponential functions can describe population growth, radioactive decay, and in our case, biological processes like heart rate scaling related to body weight.
These functions are important to understand because they:
In the allometry problem, the use of the exponent \(-1/4\) indicates an inverse relationship between heart rate and weight. As weight increases, the exponent causes the heart rate to decrease. This is a typical behavior seen when modeling natural processes where rapid changes occur. For example, exponential functions can describe population growth, radioactive decay, and in our case, biological processes like heart rate scaling related to body weight.
These functions are important to understand because they:
- Enable modeling of real-world phenomena involving rapid changes.
- Help in predicting outcomes based on changes in one or more variables.
- Are adaptable for various scientific fields, including biology and physics.
Biological Scaling Laws
Biological scaling laws describe how different biological characteristics change according to the size of organisms. These laws help explain why features such as metabolism, strength, and heart rate change as animals grow in size.
The heart rate equation given in the exercise is derived from these scaling laws. It suggests that biological functions often change in a predictable manner with size, usually following a power law as seen with the \(-1/4\) exponent. This means that larger animals tend to have slower heart rates, a pattern consistent with the mechanics of their physiology and energy use.
Scaling laws are crucial in biology because they:
The heart rate equation given in the exercise is derived from these scaling laws. It suggests that biological functions often change in a predictable manner with size, usually following a power law as seen with the \(-1/4\) exponent. This means that larger animals tend to have slower heart rates, a pattern consistent with the mechanics of their physiology and energy use.
Scaling laws are crucial in biology because they:
- Provide insights into the evolutionary adaptations of different species.
- Help bridge understanding across species by comparing their biological functions.
- Explain the relationships between size and performance among organisms.
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