Problem 53 - Algebra Form and Function

Question

The surface area, \(S\), in \(\mathrm{cm}^{2}\), of a mammal of mass \(M\) \(\mathrm{kg}\) is given by \(S=k M^{2 / 3},\) where \(k\) depends on the body shape of the mammal. For people, assume that \(k=1095 .\) (a) Find the body mass of a person whose surface area is \(21,000 \mathrm{~cm}^{2}\) (b) What does the solution to the equation \(1095 M^{2 / 3}=30,000\) represent? (c) Express \(M\) in terms of \(S\).

Step-by-Step Solution

Answer

**Answer**: The body mass of the person is about 82.78 kilograms.

Step 1: (a) Calculate the body mass of a person with the given surface area

We are given the surface area as \(S = 21,000\mathrm{cm}^2\) and the value of \(k\) for people as \(k = 1095\). The formula relating the surface area and body mass is \(S = k * M^{2/3}\). To find body mass \(M\), we can use the given values of \(S\) and \(k\) to solve for \(M\): 1. Plug in the given values: \(21,000 = 1095 * M^{2/3}\) 2. Divide by \(k\): \(M^{2/3} = \cfrac{21,000}{1095}\) 3. Calculate the result of the division: \(M^{2/3} = 19.178\) 4. To solve for \(M\), raise both sides to the power of \(\cfrac{3}{2}\) (the reciprocal of \(\cfrac{2}{3}\)): \(M = 19.178^{\frac{3}{2}}\) 5. Calculate the result: \(M \approx 82.78\) So the body mass of the person is about \(82.78\) kilograms.

Step 2: (b) Interpret the solution to the given equation

We are given the equation \(1095 * M^{2/3} = 30,000\). This equation represents a person with a certain body mass \(M\), such that their surface area is equal to 30,000 cm². The solution to this equation will give us the body mass of a person with a surface area of 30,000 cm², similar to the calculation done in part (a).

Step 3: (c) Express M in terms of S

To express \(M\) in terms of \(S\), start with the original equation, \(S = k * M^{2/3}\). We want to solve for \(M\), so we will perform the following steps: 1. Divide both sides by \(k\): \(\cfrac{S}{k} = M^{2/3}\) 2. Raise both sides to the power of \(\cfrac{3}{2}\) (the reciprocal of \(\cfrac{2}{3}\)): \(M = \left(\cfrac{S}{k}\right)^{3/2}\) Now, the mass, \(M\), is expressed in terms of surface area, \(S\): \(M = \left(\cfrac{S}{k}\right)^{3/2}\).

Key Concepts

Surface Area Calculation

Understanding how the surface area calculation relates to different physical properties is crucial. In our problem, surface area, denoted as \(S\), is calculated based on body mass \(M\) and a constant \(k\), which depends on the body's shape. The general formula is:

• \(S = k \cdot M^{2/3}\)

For humans, the constant \(k\) was provided as 1095. This constant adjusts the formula to the particular proportions of a human body.
To solve for body mass when the surface area is given, you start by isolating \(M^{2/3}\):

• Divide both sides by \(k\): \(M^{2/3} = \frac{S}{k}\)

Then, to remove the exponent \(2/3\), you raise both sides to its reciprocal, \(3/2\):

• \(M = \left(\frac{S}{k}\right)^{3/2}\)

These calculations help establish a direct relationship between the surface area of a body and its mass, offering insights into biological scaling laws.

Body Mass Formula

The body mass formula arises from biological principles where the body's surface area adapts to a power law of the mass. This type of scaling is common in nature, providing insight into how size affects other physiological functions. Here, we used the formula:

• \(M = \left(\frac{S}{k}\right)^{3/2}\)

This reveals a sophisticated relationship where body mass correlates with a fractional power of surface area. Decimal exponents often appear in biological scaling, capturing how organisms grow or change with their size.
In part (b) of the exercise, we evaluated the equation \(1095 \cdot M^{2/3} = 30,000\), which indicates a unique body mass corresponding to a specific surface area of 30,000 cm². This equation, and others like it, play essential roles in estimating when detailed physical measurements are impractical.

Exponential Equations

Exponential equations, such as the ones used in this exercise, are key in modeling numerous real-world phenomena. Though the base isn't a whole number, \(M^{2/3}\) behaves much like a standard exponential function, where instead of a constant exponent, we see fractional powers that reflect more complex relationships.
Solving these involves principles of inverse operations, such as raising to reciprocals of exponents, to determine the variable of interest. The primary strategy when dealing with exponentiation is to manipulate the equation to isolate the variable followed by inverse operations that correspond to the given exponent.
These exponential relationships underscore how subtle changes in one quantity can significantly impact another, highlighting the interconnectedness seen in natural systems and mathematical concepts alike.