Problem 93 - Applied Calculus

Question

93 -94. 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 16 -pound \(\operatorname{dog}\).

Step-by-Step Solution

Answer

The heart rate of a 16-pound dog is approximately 125 beats per minute.

Step 1: Identify Given Variables

We are given the weight of a dog as 16 pounds and an equation to calculate the heart rate: \( \text{Heart rate} = 250(\text{Weight})^{-1/4} \). The weight value we'll use in the equation is 16.

Step 2: Substitute and Set Up Equation

Substitute the given weight value into the equation for heart rate. We have: \( \text{Heart rate} = 250 \times (16)^{-1/4} \).

Step 3: Calculate the Power of Weight

Calculate \((16)^{-1/4}\). The expression \(16^{-1/4}\) means to take the fourth root of 16 and then take the reciprocal:\(16^{1/4} = \sqrt[4]{16} = 2\). Therefore, \(16^{-1/4} = 1/2\).

Step 4: Compute Heart Rate

Substitute the calculated value \(16^{-1/4} = 1/2\) back into the equation: \( \text{Heart rate} = 250 \times 1/2 = 125 \).

Step 5: Conclusion

The heart rate of a 16-pound dog is approximately 125 beats per minute.

Key Concepts

Heart Rate Calculation

Understanding how to calculate heart rates across different animal sizes can provide valuable insights, especially given that smaller animals often have faster heart rates than larger ones. This relationship is expressed using the formula:

• Heart Rate = 250(Weight)^-1/4

This calculation can be broken down into simple steps. First, we identify the given values, such as the weight of the animal (in this case, a dog's weight of 16 pounds). Then, we substitute this weight into the formula. The critical part lies in calculating the power of the weight, where we find the reciprocal of the fourth root of the weight. Once calculated, this value is multiplied by 250 to get the final heart rate in beats per minute. This modeling framework allows us to estimate the heart rate for various sizes of animals accurately.

Animal Physiology

Animal physiology explores how different physical characteristics, such as weight, influence bodily functions like heart rate. The observed trend is that smaller animals generally have higher heart rates. This is because smaller animals tend to have a higher metabolic rate per unit of body mass, requiring their hearts to beat faster to sustain their energy demands. Conversely, larger animals have slower heart rates.
This physiological principle is captured through allometric scaling, which is a biological law describing how characteristics are related to body size. The relationship given in the heart rate formula exemplifies this, showing that metabolism and heart rate are not linear with body size, but rather follow a power law. This scaling explains phenomena from tiny mice to large elephants and helps understand how anatomy and physiology adapt through evolutionary processes.

Mathematical Modeling

Mathematical modeling in animal physiology allows us to predict and understand complex biological phenomena using mathematical equations. The formula used for heart rate estimation is an example of such a model:

• It shows how we can use a simple mathematical expression to model a physiological process.
• The model generalizes the relationship between body weight and heart rate across different species.

Models like this are developed based on empirical data and aim to accurately reflect real-world scenarios. They help researchers simulate, analyze, and predict outcomes without needing direct experimental intervention. Understanding the assumptions and limitations of these models is crucial, as they simplify natural processes. Nonetheless, mathematical modeling remains a vital tool in explaining and visualizing animal behaviors and characteristics.