Problem 88 - Calculus for Biology and Medicine

Question

Adapted from Reiss, 1989\()\) In a case study in which the maximal rates of oxygen consumption (in \(\mathrm{ml} / \mathrm{s}\) ) of nine species of wild African mammals (Taylor et al., 1980 ) were plotted against body mass (in \(\mathrm{kg}\) ) on a log-log plot, it was found that the data points fell on a straight line with slope approximately equal to \(0.8\) and vertical-axis intercept approximately equal to \(0.105 .\) Find an equation that relates maximal oxygen consumption and body mass.

Step-by-Step Solution

Answer

The equation is \( y = 1.273 \cdot x^{0.8} \).

Step 1: Understanding Log-Log Plots

A log-log plot is a graph where both the x-axis (independent variable) and y-axis (dependent variable) are on a logarithmic scale. This means that a straight line on such a plot indicates a power-law relationship between the two variables.

Step 2: Identifying the Relationship

The equation of a straight line in a log-log plot can be expressed as follows: \[ \log(y) = m \log(x) + c \]where \(y\) represents maximal oxygen consumption, \(x\) represents body mass, \(m\) is the slope of the line, and \(c\) is the y-intercept.

Step 3: Substitute Known Values

We are given that the slope \(m\) is approximately \(0.8\) and the intercept \(c\) is approximately \(0.105\). Substitute these into the equation:\[ \log(y) = 0.8 \log(x) + 0.105 \]

Step 4: Convert to a Power-Law Equation

Rewriting the equation from Step 3 in terms of \(y\), we exponentiate both sides to solve for \(y\): \[ y = 10^{0.8 \log(x) + 0.105} \] Using properties of exponents, this simplifies to \[ y = 10^{0.8 \log(x)} \cdot 10^{0.105} \] which is \[ y = x^{0.8} \cdot 10^{0.105} \].

Step 5: Simplifying Further

Calculate \(10^{0.105}\) to further simplify the equation. Calculating gives approximately \(1.273\). Therefore the equation becomes:\[ y = 1.273 \cdot x^{0.8} \].

Key Concepts

Log-Log Plot

In a log-log plot, both the x and y axes are set on a logarithmic scale. This transformation allows you to explore the relationship between two variables where one or both of them change exponentially. In more simplified terms, when both variables are expressed in terms of their logs, a linear relationship will indicate a power-law relationship.

Log-log plots are particularly useful in biology and other sciences when you want to understand how two quantities scale with one another over several orders of magnitude. It helps in highlighting trends that might not be readily apparent in standard plots. For instance, in the case study with nine species of African mammals, the maximal rate of oxygen consumption and body mass were easier to analyze using a log-log plot. Through this method, a straight line was obtained, suggesting a predictable scaling pattern.

• Both axes represent logarithms of the original scales.
• If the plot shows a straight line, it implies a power-law relationship.
• Interpreting slopes and intercepts becomes easier in log-log plots, providing insights into scaling laws.

Power-Law Relationship

A power-law relationship is when one quantity varies as a power of another. It can be expressed in the form: $$ y = a \cdot x^b $$ where \( y \) changes according to the power of \( x \), \( a \) is a constant, and \( b \) is the exponent that describes the scaling relationship.
In the context of this exercise, the log-log plot revealed that maximal oxygen consumption in mammals scales with body mass through a power-law relationship. The slope of the line in a log-log plot gives the exponent \( b \), and the intercept provides information about the constant \( a \) after transforming back from logs.

• Power-law relationships are linear in log-log plots.
• Such relationships feature prominently in biology, indicating how traits scale across species.
• The slope of the log-log plot translates directly to the power exponent.

Equation Derivation

Deriving an equation from a log-log plot involves a few straightforward steps to integrate known parameters into the power-law form.
First, by finding both the slope and y-intercept from the log-log plot, we get a linear equation like: \[ \log(y) = m \cdot \log(x) + c \] where \( m \) is the slope and \( c \) is the intercept on the y-axis.
Substitute the given values of the slope (0.8) and intercept (0.105) to form:\[ \log(y) = 0.8 \cdot \log(x) + 0.105 \] Next, solve for \( y \) by exponentiating both sides of the equation, which reverses the logarithmic transformation:\[ y = 10^{0.8 \cdot \log(x) + 0.105} \]This can be restructured using properties of exponents as:\[ y = x^{0.8} \cdot 10^{0.105} \]Finally, calculating \( 10^{0.105} \) gives approximately 1.273, leading to the simplified formula:\[ y = 1.273 \cdot x^{0.8} \]This equation effectively describes how maximal oxygen consumption scales with body mass across the studied mammal species.