Problem 14 - Applied 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 \(E = k \cdot v^3\), where \(k\) is a constant.

Step 1: Understand the Proportionality

We are given that the energy, \(E\), is proportional to the cube of the speed, \(v\). This means that if you increase \(v\), \(E\) will increase by the cube of \(v\). We denote this relationship using a proportionality constant \(k\).

Step 2: Write the Proportionality Equation

Since \(E\) is proportional to \(v^3\), we can write this relationship as \(E = k \cdot v^3\), where \(k\) is the constant of proportionality.

Step 3: Interpret the Formula

The formula \(E = k \cdot v^3\) expresses that the energy expended by the dolphin increases as the cube of its speed. This means doubling the speed will increase the energy expended by a factor of eight because \((2)^3=8\).

Key Concepts

Energy and Work

The relationship between energy and work is a fundamental concept in physics. Energy is defined as the capacity to do work. Work, in a scientific context, is performed when a force causes an object to move. This relationship is critical when understanding how energy is expended or transferred in various scenarios.
For the swimming dolphin, energy is expended against the resistance of water. The energy needed increases depending on the dolphin's speed. Since energy in this case relates to the dolphin's ability to overcome water resistance, it is closely linked to the concept of work.
In the formula provided, where energy expended, \(E\), is proportional to the cube of speed, \(v\), we understand that the work done by the dolphin involves pushing water at increasingly larger scales as the speed increases. Hence, the work relates to how much energy must be expended to maintain or increase speed, illustrating why the cube relation fits nature's observation.

Speed and Velocity

Speed and velocity are key concepts often discussed in physics and essential in understanding motion. Speed is a scalar quantity, referring solely to how fast an object moves regardless of direction. Velocity, however, is a vector, incorporating both the speed and direction of the moving object.
When examining the energy expended by a swimming dolphin, we refer to speed, specifically how quickly the dolphin moves through water without consideration of direction. This simplifies to a scalar quantity that is fundamental to calculating energy in relation to speed. When we say the energy is proportional to the cube of speed, changes in velocity's direction don't affect the energy calculation unless it changes the speed component.

• If the speed of the dolphin doubles, the energy expenditure increases eightfold due to the cube relationship, reflecting a significant increase in the dolphin’s effort.
• This insight highlights how speed directly impacts energy consumption, guiding us to consider efficient movement both in nature and technology.

Mathematical Formulas

Mathematical formulas serve as the language of science that quantifies relationships observed in the physical world. In this scenario, the formula \(E = k \cdot v^3\) is a concise representation of how energy relates to speed. Each variable and constant in the formula holds specific scientific meaning.
The symbol \(E\) represents energy expended, while \(v\) is the speed of the dolphin. The constant \(k\) is crucial as it adjusts the formula to match real-world conditions, accounting for factors like water resistance and dolphin physiology.
Understanding formulas like this one helps deepen comprehension of natural phenomena and aids in predicting outcomes. For instance, knowing the cube relationship empowers us to accurately anticipate energy needs based on speed changes, a practical application in fields like biomechanics and marine biology. The elegance of mathematical formulas lies in their universality and ability to convey complex interactions in digestible expressions.