Problem 31 - Calculus

Question

Write a formula representing the function. The number of animal species, \(N,\) of a certain body length, \(l\), is inversely proportional to the square of \(l\)

Step-by-Step Solution

Answer

The function is \( N = \frac{k}{l^2} \).

Step 1: Understand the concept of inverse proportionality

If a quantity is inversely proportional to another, it means that as one quantity increases, the other decreases at a specific rate. In this problem, the number of animal species, \(N\), is inversely proportional to the square of the body length \(l\).

Step 2: Define the inverse proportionality relationship

The relationship can be expressed mathematically as \( N \propto \frac{1}{l^2} \). This equation indicates that \(N\) is proportional to the inverse of the square of \(l\).

Step 3: Introduce the constant of proportionality

To turn the proportionality into an equation, introduce a constant \(k\). This gives us the equation \( N = \frac{k}{l^2} \), where \(k\) is a constant that determines the specific nature of the relationship between \(N\) and \(l^2\).

Step 4: Write the final formula for the function

The formula representing the function, considering the given inverse proportionality, is: \[ N = \frac{k}{l^2} \]. This equation can be used to find the number of species \(N\) for a given body length \(l\), provided the constant \(k\) is known.

Key Concepts

Square of a Variable

When we talk about the "square of a variable," we are referring to multiplying the variable by itself. So, if we have a variable represented by the letter \(l\), the square of \(l\) is written as \(l^2\), which equals \(l \times l\).
This squared term is important in mathematical equations because it shows how a quantity grows exponentially rather than linearly. For example, if \(l\) represents a body length of an animal in meters, \(l^2\) represents the area that gets covered as this length increases.
In the context of our problem, \(N \propto \frac{1}{l^2}\) indicates that the number of species, \(N\), is influenced by the area (\(l^2\)) calculated from the animal's length \(l\). Understanding squaring is key to working with relationships involving areas or any other squared units.

Constant of Proportionality

The term "constant of proportionality" is used in mathematics to express a fixed number that describes how two variables are related to each other. When two quantities are proportional, the constant tells us how much one quantity changes when the other changes.
Imagine you know \(N = \frac{k}{l^2}\). Here, \(k\) is the constant of proportionality. This constant helps to specify the rate or strength of the inverse proportional relationship between \(N\) and \(l^2\).

• It ensures the equation is accurate by balancing both sides.
• To use \(k\), we need precise data or measurements about the specific scenario we're dealing with.

Once the value of \(k\) is known, it allows us to make exact predictions about \(N\) given different values of \(l\). Thus, this constant turns a proportional relationship into a precise equation that can be used practically.

Mathematical Relationship

A mathematical relationship describes how two or more quantities are connected. In the context of our example, the relationship between the number of species \(N\) and the body length \(l\) is described by inverse proportionality to the square of \(l\) as shown in \(N = \frac{k}{l^2}\).
This relationship gives a behavior insight: as \(l\) increases, \(N\) decreases. It's as if the size of an ecosystem can only support fewer larger animals but more smaller ones.

• It's crucial for understanding how variables in nature may affect one another.
• Mathematical relationships like this one help predict and analyze real-world situations mathematically.

Through inverse proportionality, we deduce that doubling the body length results in \(N\) becoming a quarter of its previous value, showcasing this profound and insightful mathematical linkage.