Problem 22 - 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 formula is \( B(M) = 0.05M \), and the blood mass of a 70 kg human is 3.5 kg.

Step 1: Understanding the Proportionality

We know that the blood mass of a mammal is proportional to its body mass. This means if \( M \) is the body mass and \( B \) is the blood mass, then \( B = kM \), where \( k \) is the proportionality constant.

Step 2: Finding the Proportionality Constant

We know from the problem statement that when the body mass \( M = 3000 \) kg, the blood mass \( B = 150 \) kg. Substitute these values into the equation \( B = kM \) to find \( k \).\[ 150 = k imes 3000 \]Solving this gives \[ k = \frac{150}{3000} = 0.05 \].

Step 3: Formulating the Blood Mass Equation

Now that we have \( k = 0.05 \), we can form the equation for blood mass as a function of body mass: \[ B(M) = 0.05M \]

Step 4: Estimating the Human Blood Mass

Using the formula \( B(M) = 0.05M \), substitute \( M = 70 \) kg to find the blood mass for a human. \[ B(70) = 0.05 imes 70 = 3.5 \]So, the blood mass of a human with body mass 70 kg is 3.5 kg.

Key Concepts

Body Mass

Body mass refers to the total weight of an organism, typically measured in kilograms or pounds. For mammals, body mass can vary widely between different species, ranging from very small animals like mice to massive ones like elephants. Understanding the body mass of a mammal is crucial, as it influences various physiological functions and characteristics, including metabolic rate, dietary needs, and even blood mass. When calculating the body mass of an organism, it's important to consider factors such as:

• Bone density
• Muscle and fat composition
• Overall size and structure

In the context of proportional relationships, body mass acts as a key variable to determine other related masses, like blood mass, by using a constant factor.

Blood Mass

Blood mass is the total weight of blood within a mammal's body. Blood plays a vital role in transporting oxygen, nutrients, and waste products, making its mass a critical parameter for the overall health and functionality of the organism. In proportional relationships, we try to understand how blood mass relates to other variables, such as body mass.Typically, blood mass is directly proportional to body mass. This means that as the body mass increases, the blood mass tends to increase at a consistent rate. The formula to describe this relationship is given by:\[ B = kM \]Here:

• \( B \) is the blood mass,
• \( M \) is the body mass,
• \( k \) is a constant of proportionality.

Understanding this relationship helps in estimating the blood volume needed for medical treatments and for studying evolutionary biology across different species.

Proportionality Constant

The proportionality constant, represented as \( k \) in equations, is a key element in determining how two quantities are related through direct proportionality. When one quantity changes, the other changes at a rate dictated by this constant. For the blood-to-body mass relationship in mammals, the proportionality constant can be found by using known values of body and blood mass, such as those provided by measurements from specific animals.For example, given a rhinoceros with a body mass of 3000 kg and a blood mass of 150 kg, we calculate the constant \( k \) as follows:\[ 150 = k \times 3000 \]Solving for \( k \), we find that:\[ k = \frac{150}{3000} = 0.05 \]This constant indicates that for every kilogram of body mass in a mammal, the blood mass is about 5% or \( 0.05 \times \) of that body mass. Using this constant, it becomes easy to estimate the blood mass for any given body mass, providing a simple yet powerful tool for scientific study and practical applications such as veterinary medicine.