Problem 91 - College Algebra

Question

We saw that the percentage of people satisfied with their lives remains relatively constant for all age groups. Exercise 69 showed that the number of skiers in the United States has remained relatively constant over time. Give another example of a real-world phenomenon that has remained relatively constant. Try writing an equation that models this phenomenon.

Step-by-Step Solution

Answer

An example of a constant real-world phenomenon is the speed of light in a vacuum. The corresponding mathematical model for this constant phenomenon is \(c = 299792\) km/s.

Step 1: Identify Constant Real-World Phenomenon

One example of a real-world phenomenon that has remained relatively constant is the speed of light in a vacuum. In scientific terms, the speed of light in a vacuum is universally recognized as a constant, equal to approximately \(299,792\) kilometers per second.

Step 2: Formulate Mathematical Model

An equation that models this phenomenon is \(c = 299792\) km/s. Where 'c' represents the speed of light.

Step 3: Explain the Model

This equation means that no matter the situation or observation, the speed of light in a vacuum is always \(299792\) kilometers per second, thus demonstrating a real constant phenomenon.

Key Concepts

Constants in Real-World Phenomena

In understanding the world around us, certain values remain unchanged and are fundamental to our interpretation of the natural and scientific world. These values, known as constants, play a crucial role in mathematical equations and models. A constant is a quantity that remains fixed throughout a particular context, and it does not change regardless of the situation.

For example, in physics, the gravitational constant represents the force of attraction between two objects; it is vital for calculations ranging from the trajectory of a ball to the motion of planets. In chemistry, Avogadro's number is a constant that details the number of particles found in one mole of a substance, underpinning stoichiometry and molecular interactions. Constants offer a touchstone for scientists and mathematicians, enabling them to predict outcomes and understand relationships within various systems. They serve as a foundation from which variability and change can be measured and assessed.

Speed of Light

The speed of light is perhaps one of the most famous constants, not just in physics, but in all of science. Denoted by the symbol 'c', the speed of light in a vacuum is exactly 299,792 kilometers per second (or about 186,282 miles per second). This speed is significant because it represents the maximum speed at which all energy, matter, and information in the universe can travel.

It is a fundamental constant in the theory of relativity, helping us understand the structure of space-time and the link between space and time. The constancy of the speed of light underpins many technologies we use daily, such as GPS systems, which rely on accurate timing measurement, where the speed of light comes into play. This constant also forms the basis of our understanding of the universe, from measuring astronomical distances to theorizing about the possibilities of time travel.

Mathematical Modeling

Mathematical modeling is the process of using mathematics to represent, analyze, and make predictions about real-world systems. It involves creating a mathematical representation, or model, of a phenomenon, using variables to represent different aspects of the system, and applying equations to simulate and study behaviors. Models can range from simple linear equations for predicting trends to complex differential equations that describe dynamic systems.

The beauty of mathematical modeling lies in its ability to simplify complicated realities into a form where calculations can be made to forecast or understand phenomena. It's utilized across numerous fields, including physics, engineering, economics, and even social sciences. The validity of a model is determined by how accurately it predicts the behavior of the system it represents. Over time, models can be refined with additional data and improved computational techniques to better mirror the complexities of the world.