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 \(E = k v^3\), where \(k\) is a constant.

Step 1: Understand the Problem

We are given that the energy \(E\) expended by a dolphin is proportional to the cube of its speed \(v\). We need to express this relationship as a mathematical formula.

Step 2: Identify Key Terms

The term 'proportional' in mathematics indicates that one quantity is equal to a constant multiplied by another quantity. Here, \(E\) is proportional to \(v^3\). Let \(k\) be the constant of proportionality.

Step 3: Formulate the Equation

According to the information, \(E\) is proportional to \(v^3\). Thus, the equation can be expressed as:\[E = k v^3\]Where \(k\) is the proportionality constant.

Key Concepts

Energy Expenditure

Energy expenditure refers to the amount of energy a living organism uses to perform any activity, like a dolphin swimming. This energy is not always used at a constant rate. Factors such as speed, distance traveled, and environmental conditions can affect energy use. For dolphins, energy is expended primarily through muscle activity during swimming. This energy comes from food the dolphin eats, which is converted into usable fuel within its body. The more active the dolphin, the more energy it requires. Understanding energy expenditure is important. It helps in determining the needs of organisms in their natural habitats. By studying the energy spent by dolphins at different speeds, we can learn about their feeding habits and how they optimize their energy use in oceanic environments.

Mathematical Formula

In mathematics, formulas help us represent complex relationships in simple terms. In this case, we describe the relationship between energy and speed using a formula.The relationship given in the exercise indicates proportionality. This means one quantity changes as another changes, in specific ways. For dolphins, the energy (E) they expend is proportional to the cube of their speed (v). Let's break it down:

• Proportionality: When two quantities are proportional, one equals a constant times the other.
• Cube of Speed: Cubing the speed means multiplying the speed by itself three times, which denotes a rapid increase in energy use as the speed increases.

Hence, the mathematical formula is expressed as:\[E = k v^3\]where E represents energy, v is the speed, and \(k\) is the constant of proportionality, representing other factors affecting energy besides speed.

Speed and Energy

Speed and energy have an intricate relationship, especially in the context of moving objects like dolphins. Here, the speed of the dolphin has a direct impact on the energy it uses.When we talk about the cube of speed, it means energy use does not just increase linearly, but at an accelerating rate. For instance, if a dolphin doubles its speed, the energy expended increases eight times, because \(2^3 = 8\).This principle helps us understand why faster speeds are more taxing and hence less sustainable for long durations without rest. It also emphasizes the efficiency of energy use at different speed levels. Understanding this relationship is critical in fields like wildlife research, bioenergetics, and conservation. It highlights the balance required for marine life to survive and thrive within their ecosystems.