Problem 18 - Applied Calculus

Question

The surface area of a mammal, \(S,\) satisfies the equation \(S=k M^{2 / 3},\) where \(M\) is the body mass, and the constant of proportionality \(k\) depends on the body shape of the mammal. A human of body mass 70 kilograms has surface area \(18,600 \mathrm{cm}^{2} .\) Find the constant of proportionality for humans. Find the surface area of a human with body mass 60 kilograms.

Step-by-Step Solution

Answer

The constant \(k\) is approximately 1086.67. The surface area for a 60 kg human is approximately 17,251 cm².

Step 1: Understand the Given Problem

The problem involves finding two quantities related to the surface area formula, \(S = k M^{2/3}\), where \(S\) is the surface area, \(M\) is the body mass and \(k\) is a constant. First, we need to find the value of \(k\) for a human with known mass and surface area. Then, we will use \(k\) to find the surface area for another given mass.

Step 2: Find the Constant of Proportionality (k)

Given that a human of body mass 70 kg has a surface area of 18,600 cm², plug these values into the equation \(S = k M^{2/3}\) to solve for \(k\): \[18600 = k \times 70^{2/3}\]First, calculate \(70^{2/3}\). Using a calculator, \(70^{2/3} \approx 17.1229\).Then solve for \(k\) by dividing both sides by \(17.1229\): \[k = \frac{18600}{17.1229} \approx 1086.67\]

Step 3: Use the Constant to Find Surface Area for 60 kg Mass

Now that we know \(k \approx 1086.67\), use this value to find the surface area for a person with a mass of 60 kg using the formula: \[S = k \times 60^{2/3}\] First, calculate \(60^{2/3}\). Using a calculator, \(60^{2/3} \approx 15.874\). Then calculate the surface area, \(S\):\[S = 1086.67 \times 15.874 \approx 17250.97\]So the surface area is approximately \(17250.97\) cm².

Key Concepts

Body Mass

Body mass is a critical factor in determining the surface area of an organism, particularly in mammals like humans. When we talk about body mass, we are referring to the total amount of matter contained in an individual's body, measured typically in kilograms or pounds. In the context of our exercise, body mass is crucial because it directly influences the surface area via the equation, \(S = k M^{2/3}\).

• "\(M\)" represents the body mass in this surface area equation.
• The value of \(M\) is raised to the power of \(2/3\), demonstrating that surface area does not increase linearly with mass.

By understanding body mass, one can appreciate how small changes in weight can lead to variations in the calculated surface area. This concept is essential for applications ranging from biological studies to engineering human-suited objects.

Proportionality Constant

The proportionality constant, denoted as \(k\), plays a pivotal role in the mathematical equation \(S = k M^{2/3}\). It essentially bridges the relationship between body mass and surface area.

• The value of \(k\) is specific to the shape and characteristics of the mammal's body.
• Finding \(k\) involves reorganizing the equation and solving for the constant when provided with empirical data, such as mass and surface area.

In our example, by applying the given mass and surface area for a person, we determined the constant to be approximately \(1086.67\). This constant can now be used as a reliable factor when predicting the surface area of humans within the same range of body mass.

Mathematical Modeling

Mathematical modeling is a powerful technique used to represent real-life scenarios through mathematical expressions. In this exercise, the formula \(S = k M^{2/3}\) serves as a model describing how surface area relates to body mass.

• The equation creates a simplified version of the relationship between surface area and mass.
• By utilizing this model, predictions about surface area can be made for different masses, assuming the same proportionality constant applies.

Models like this are crucial for scientific research, engineering, and various fields where understanding complex systems in a simplified manner is imperative. They allow us to predict, analyze, and optimize without the need for exhaustive empirical testing.