Problem 30 - Calculus

Question

Write a formula representing the function. The energy, \(E,\) expended by a swimming dolphin is proportional to the cube of the speed, \(v,\) of the dolphin.

Step-by-Step Solution

Answer

The formula is \(E = k \cdot v^3\).

Step 1: Understand Proportionality

The problem states that the energy \(E\) is proportional to the cube of the speed \(v\). Proportionality here means that when one quantity changes, the other changes in a fixed ratio.

Step 2: Express Proportional Relation in Formula

To express the proportion mathematically, we use \(E \propto v^3\), where \( \propto \) denotes 'is proportional to'. This means \(E\) can be written as \(E = k \cdot v^3\) where \(k\) is the constant of proportionality.

Step 3: Define the Formula

The complete formula is \(E = k \cdot v^3\), indicating the relationship between energy expended and the cube of the speed. \(k\) must be determined through additional information, such as specific measurements or experiments.

Key Concepts

Energy and Speed Relationship

The relationship between energy and speed is a fascinating topic. In the scenario of a swimming dolphin, the energy expended is directly linked to its speed. Specifically, energy, denoted as \( E \), is proportional to the cube of the dolphin's speed, represented by \( v \). This means that if the dolphin swims faster, the energy it uses increases significantly. To visualize this, consider if the dolphin doubles its speed. Given that energy is proportional to the cube of the speed, the energy will increase by \( 2^3 \) or 8 times. This relationship highlights how significant the increase in energy cost can be due to a change in speed. This concept is critical in understanding energy dynamics in movements of not just dolphins, but other fast-moving creatures as well. This relationship shows how energetically costly faster speeds can be and provides insight into how dolphins, and animals in general, balance speed with energy conservation.

Proportionality Constant

The proportionality constant, often denoted as \( k \), plays an essential role in the equation \( E = k \cdot v^3 \). This constant provides a fixed ratio linking energy and the cube of speed. Without this constant, the relationship would remain undefined, as \( k \) adjusts the mathematical model to represent real-world behavior accurately.The constant \( k \) is crucial when creating the formula. It can vary significantly depending on various factors, such as the animal in question or the medium through which it moves (like water or air). Obtaining the value of \( k \) often requires practical measurements or experimentation, which means running tests under controlled conditions to determine energy expenditure at different speeds. By knowing the value of \( k \), it becomes possible to predict how much energy the dolphin will need as its speed changes, thereby facilitating better understanding and planning around its movement and energy consumption.

Mathematical Modeling

Mathematical modeling is a powerful tool that allows us to represent real-world phenomena using mathematical expressions. In this case, the model \( E = k \cdot v^3 \) represents the energy used by a dolphin as a function of its speed. Such models help simplify complex relationships into understandable formulas that can be easily applied and manipulated.Creating a mathematical model involves identifying key variables and establishing their relationships. Here, the primary variables are energy \( E \) and speed \( v \), with a critical relationship governed by the cube of speed and a proportionality constant \( k \). This model aids in understanding how altering speed affects energy consumption.Using mathematical models allows scientists and engineers to predict outcomes and optimize performance in various fields. For instance, in biology, it can describe energy use in animals, while in physics, it can help design energy-efficient systems. Thus, such models form the backbone of solving many real-world problems efficiently.