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 840.43. For a 60 kg human, the surface area is about 15,316.53 cm².

Step 1: Identify Variables and Known Quantities

The problem states that the surface area \( S \) is given by the equation \( S = k M^{2/3} \), where \( M \) is the body mass and \( k \) is the constant of proportionality. We know from the problem that for a human with a body mass of \( 70 \text{ kg} \), the surface area \( S \) is \( 18,600 \text{ cm}^2 \). Our task is to find \( k \).

Step 2: Substitute Known Values into the Equation

Substitute \( M = 70 \) kg and \( S = 18,600 \text{ cm}^2 \) into the equation \( S = k M^{2/3} \). This gives:\[18,600 = k (70)^{2/3}\]

Step 3: Solve for the Constant of Proportionality \( k \)

To isolate \( k \), first calculate \( 70^{2/3} \). This gives us about \( 22.14 \) (rounded to two decimal places). Now solve for \( k \):\[k = \frac{18,600}{22.14}\]\( k \approx 840.43 \).

Step 4: Calculate Surface Area for Second Body Mass

Now, use the constant \( k \approx 840.43 \) to find the surface area for a human with a body mass of \( 60 \text{ kg} \). Use the equation:\[S = k M^{2/3}\]Substitute \( k = 840.43 \) and \( M = 60 \):\[S = 840.43 (60)^{2/3}\]

Step 5: Compute the New Surface Area

Calculate \( 60^{2/3} \), giving around \( 18.23 \). Plug this into the equation for \( S \):\[S = 840.43 \times 18.23 \approx 15,316.53 \text{ cm}^2\]

Key Concepts

Constant of Proportionality

In mathematics, the constant of proportionality is a value that relates two quantities that are directly proportional. In our exercise, the surface area \( S \) and the body mass \( M \) of a mammal are connected using the equation \( S = k M^{2/3} \). The constant \( k \) is crucial as it adjusts the equation to fit the specific characteristics of the object being measured, such as the shape and density of the mammal.

• **Finding \( k \):** To find \( k \), we took a known example, that of a human with a surface area of \( 18,600 \text{ cm}^2 \) and body mass of \( 70 \text{ kg} \). By substituting these into the equation, \( k \) was calculated as approximately \( 840.43 \). This value essentially scales the relationship to match the physical reality for humans.
  
• **Significance:** Without the appropriate \( k \), the equation wouldn’t accurately represent the surface area across different mammals or different body masses within the same species. It's a pivotal factor for biological calculations.
  

Body Mass

Body mass is a measure of the amount of matter in an object, typically measured in kilograms or grams. It has a significant biological influence on many factors, including metabolic rates and surface area.

• **Role in the Equation:** In the equation \( S = k M^{2/3} \), body mass \( M \) is raised to the power of \( \/3 \) which implies a direct but non-linear relationship with the surface area. This indicates that as body mass changes, the surface area changes at a less than proportional rate.
  
• **Practical Calculation:** Let's take the example where \( M = 60 \text{ kg} \) and using the previously calculated \( k \): The surface area is \( S = 840.43 \times (60)^{2/3} \approx 15,316.53 \text{ cm}^2 \). The body mass is key to predicting the surface area efficiently.
  

Applied Calculus

Applied calculus is a branch of mathematics that involves using calculus concepts to solve real-world problems. In our context, it helps in deriving equations that describe complex relationships between biological quantities.

• **Usage in Surface Area Calculation:** Calculus aids in understanding how surface area, a nonlinear function of body mass in this case, behaves as the mass changes. This is captured in the power \( M^{2/3} \), which requires calculus to understand its derivation and implications.
  
• **Importance:** Applied calculus not only provides the tools to calculate the surface area but also helps to frame biological phenomena in mathematical terms, making it easier to interrelate and infer physiological aspects.
  
• **Approach:** While solving the problem, the use of derivatives could further benefit in understanding how slight changes in mass result in changes in the surface area. However, our current task was more algebraic, focusing on substituting values and solving the equation for specific numbers.