Problem 20 - Applied Calculus

Question

The blood mass of a mammal is proportional to its body mass. A rhinoceros with body mass 3000 kilograms has blood mass of 150 kilograms. Find a formula for the blood mass of a mammal as a function of the body mass and estimate the blood mass of a human with body mass 70 kilograms.

Step-by-Step Solution

Answer

The estimated blood mass of a 70 kg human is 3.5 kg.

Step 1: Understand the Proportionality

The problem states that the blood mass of a mammal is proportional to its body mass. This means if we denote the body mass by \( M_b \) and the blood mass by \( M_{bl} \), then \( M_{bl} = k \cdot M_b \), where \( k \) is the proportionality constant.

Step 2: Find the Proportionality Constant

Using the given information about the rhinoceros, we have \( M_b = 3000 \) kg and \( M_{bl} = 150 \) kg. Substituting into the equation \( M_{bl} = k \cdot M_b \), we get:\[ 150 = k \cdot 3000 \]Solving for \( k \), we divide both sides by 3000:\[ k = \frac{150}{3000} = 0.05 \]

Step 3: Derive the Blood Mass Formula

Substitute the value of \( k \) back into the proportionality equation to get the formula for blood mass as a function of body mass:\[ M_{bl} = 0.05 \cdot M_b \]

Step 4: Estimate Blood Mass for a Human

To estimate the blood mass of a human with a body mass of 70 kg, use the formula derived:\[ M_{bl} = 0.05 \cdot 70 = 3.5 \]

Step 5: Conclusion

The estimated blood mass of a human with a body mass of 70 kilograms is 3.5 kilograms.

Key Concepts

Blood Mass

Blood mass refers to the total weight of blood contained in the body of a mammal. Blood plays a crucial role in transporting nutrients, oxygen, and hormones to the cells, as well as removing waste products. The mass of blood is often a function of a mammal's body mass. This means that larger animals generally have more blood, while smaller species tend to have less. In problems of proportionality, the relationship between blood mass and body mass is generally linear. The equation can be expressed as:

• \( M_{bl} = k \cdot M_b \)

where \( M_{bl} \) is the blood mass, \( M_b \) is the body mass, and \( k \) is a constant of proportionality. For instance, in the given exercise, the rhinoceros has a blood mass proportional to its body mass, modeled by the formula \( M_{bl} = 0.05 \cdot M_b \). This formula means that for every 100 kilograms of body weight, the blood mass is 5 kilograms. By using similar calculations, one can determine the blood mass for different mammals given their body mass.

Body Mass

Body mass is the total weight of a mammal, encompassing all its tissues, organs, and fluids. It is a key biological parameter that influences various physiological functions of organisms. In the context of mammals, body mass can provide insights into their overall health, energy requirements, and metabolism. For a rhinoceros, a substantial body mass of 3000 kg implies the need for a relatively larger volume of blood to supply oxygen and nutrients throughout the body. In contrast, a human or smaller mammal will have a significantly lower body mass and consequently, lower blood volume requirements.

• The relation between body mass and other physiological parameters, such as blood mass, helps understand how mammals adapt to their environments and manage their energy needs.
• These relations can be particularly useful in ecological and veterinary sciences for estimating the health and nutritional needs of various species based on simple body mass measurements.

Understanding the dynamics of body mass can help in creating models like the one used in the rhino example, where the blood mass is directly proportional to body mass, aiding in the estimation of such biological features across different mammals.

Mammals

Mammals are a diverse class of animals known for their warm-blooded nature and the presence of fur or hair, as well as mammary glands which females use to nourish their young. This group includes a broad range of species, from the tiniest rodents to the largest whales. Despite their diversity, many mammals share common physiological traits, including a circulatory system that is proportional to their body mass.

• Mammals' blood circulates via a closed circulatory system, which is pivotal to maintaining their active and high-energy lifestyles.
• A mammal's body mass and blood mass relationship reflects its evolutionary adaptations to be efficient, ensuring adequate blood supply reaches all vital organs.

By understanding how mammals balance blood mass with body mass, researchers and scientists can gain insights into how these animals function and thrive. For example, smaller mammals often have higher metabolic rates compared to larger ones, necessitating a greater relative blood supply. The proportionality constant, like the one derived in the rhino example, helps in understanding these relationships and can be applied to various ecological and biological studies across different environments and climates.