Problem 23 - Applied Calculus

Question

Biologists estimate that the number of animal species of a certain body length is inversely proportional to the square of the body length. \(^{88}\) Write a formula for the number of animal species, \(N,\) of a certain body length as a function of the length, \(L .\) Are there more species at large lengths or at small lengths? Explain.

Step-by-Step Solution

Answer

There are more species at smaller lengths because \( N = \frac{k}{L^2} \) increases as \( L \) decreases.

Step 1: Understand Inverse Proportionality

When a quantity is said to be inversely proportional to another quantity, it means that as one increases, the other decreases. If the number of animal species \( N \) is inversely proportional to the square of body length \( L \), then this relationship can be expressed as \( N \propto \frac{1}{L^2} \).

Step 2: Establish a Formula

From inverse proportionality, we can introduce a constant of proportionality, \( k \), leading to the equation: \( N = \frac{k}{L^2} \). Here, \( k \) is a positive constant that depends on other factors like the specific ecological context that aren't detailed in the problem.

Step 3: Analyze the Relationship

Now that we have the formula \( N = \frac{k}{L^2} \), observe that as \( L \) increases, \( L^2 \) also increases, and thus \( \frac{k}{L^2} \) decreases. Conversely, as \( L \) decreases, \( \frac{k}{L^2} \) increases.

Step 4: Conclude about Species Distribution

Given the inverse relationship in the formula \( N = \frac{k}{L^2} \), more animal species exist at smaller body lengths because \( N \) becomes larger as \( L \) decreases.

Key Concepts

Number of Animal Species

The term "number of animal species" refers to the total count of different species within a specific size range, particularly focusing on those with similar body lengths. When biologists study species diversity, they often measure how many different species exist at various body lengths, providing insights into ecological richness and diversity.
Understanding the number of animal species at particular body lengths helps researchers identify patterns of biodiversity. For example:

• Animals with shorter body lengths might be more numerous for evolutionary or environmental reasons.
• This reflects adaptability and availability of ecological niches for smaller animals.

In our problem, the formula derived shows that smaller animals are indeed more common, illustrated by the increase in the number of species as the body length decreases.

Body Length

Body length, in this context, is the measurable size aspect of an animal from one end to the other. It's a crucial factor that influences which animals thrive or diminish in numbers.
Various factors such as habitat space, resources, and predation risks can affect body length distributions among animal species.

• Smaller animals might have advantages like requiring fewer resources or being more agile, increasing survival rate.
• Larger animals may face challenges like needing more resources, which could limit their population numbers under certain conditions.

In inverse proportionality, body length affects the number of species, with shorter lengths typically supporting a higher number of species, as seen in the described relationship.

Formula Derivation

Deriving a formula in mathematics helps translate real-world phenomena into expressions that can be easily analyzed and understood. In our exercise, we derived a formula representing the relationship between the number of species and body length:First, we introduced inverse proportionality. This means that the number of species, \( N \), is inversely proportional to the square of body length, \( L \). Mathematically, this is expressed as, \( N \propto \frac{1}{L^2} \).To refine this into an equation that is more useful in calculations, we introduce a constant \( k \). Now, we have:\[ N = \frac{k}{L^2} \]This equation tells us that as \( L \) (the body length) becomes larger, \( N \) (the number of species) becomes smaller. Conversely, smaller body lengths correspond with a larger number of species. Deriving such formulas provides a meaningful way to predict and understand the natural distribution of animal species relative to their body lengths.