Question

Independent and Dependent Variables Identify which of the two variables in Exercises \(10-14\) is the independent variable \(x\) and which is the dependent variable \(y .\)Speed of a wind turbine and the amount of electricity generated by the turbine.

Short answer

Answer: The independent variable is the speed of a wind turbine, and the dependent variable is the amount of electricity generated by the turbine.

Step-by-step solution

Identify the variables.

There are two variables in this exercise: 1. Speed of the wind turbine 2. Amount of electricity generated by the turbine

Determine which variable depends on the other

We need to figure out how the variables are related. In this case, the amount of electricity generated by the turbine depends on the speed of the wind turbine. When the speed of the wind turbine increases, the amount of electricity generated by the turbine generally increases.

Assign the independent and dependent variables.

Now that we know how the variables are related, we can assign the independent variable \(x\) and the dependent variable \(y\). - The independent variable \(x\) is the speed of the wind turbine, as it does not depend on the amount of electricity generated. - The dependent variable \(y\) is the amount of electricity generated by the turbine since it depends on the speed of the wind turbine. So the independent variable \(x\) is the speed of a wind turbine and the dependent variable \(y\) is the amount of electricity generated by the turbine.

Key concepts

Wind Turbine Speed

Understanding wind turbine speed is essential in the field of renewable energy. A wind turbine operates by converting the kinetic energy of wind into mechanical energy, which is then turned into electricity. The speed at which the turbine blades spin is known as the wind turbine's rotational speed, which is typically measured in rotations per minute (RPM).

The efficiency with which a wind turbine converts wind energy into electricity is highly influenced by its speed. When the wind blows faster, the blades rotate more swiftly, harnessing more energy that can be converted into electricity. It's not a linear relationship, though: turbines are designed to operate within an optimal speed range and will shut down to prevent damage if the wind is too strong.

Additionally, modern wind turbines are equipped with intelligent control systems to adjust the blade pitch, thus maximizing efficiency over varying wind speeds. These systems make it possible to maintain a consistent electrical output even when wind speeds fluctuate. Understanding the independent variable of wind turbine speed is key for optimizing energy production and ensuring the longevity of the equipment.

Electricity Generation

Electricity generation via wind turbines is a cornerstone of sustainable energy initiatives. The process begins as wind energy forces the turbine blades to rotate, driving a shaft connected to a generator. This mechanical motion is then converted into electrical energy through electromagnetic induction.

The amount of electricity generated is dependent on several factors, including the wind turbine's speed. Higher wind speeds yield more blade rotations per minute, producing more electricity up until the turbine reaches its safety limits. This is the core of understanding the dependent variable—electricity output relies on the speed at which the wind is turning the blades.

Different types of turbines, such as onshore and offshore models, have varying capabilities and efficiencies. Moreover, the design and technology of the turbine, such as the number of blades and the materials used for construction, also affect electricity generation. These machines are carefully engineered to convert as much wind energy into electricity as possible, catering to our increasing demand for renewable energy sources.

Variable Relationships

The relationship between variables in any scientific experiment or real-world situation is critical for data analysis and understanding system behaviors. In our context, the independent variable is the wind turbine speed, and the dependent variable is the electricity generation. The relationship here indicates that as the speed at which the turbine blades spin changes, the electricity produced also changes accordingly.

This relationship can be described as a direct relationship, where an increase in one variable leads to an increase in the corresponding dependent variable, given that other factors remain constant. The study of these relationships is fundamental in fields such as physics, engineering, and economics because it allows for the prediction and manipulation of outcomes based on changes in input variables.

To visualize and analyze variable relationships, scientists and engineers use various methods, including graphing data points on a coordinate system where the independent variable (x-axis) and the dependent variable (y-axis) can be plotted and compared. Understanding the nature of these relationships is essential not just for predicting outcomes, but also for designing systems that can optimize specific variables to achieve desired results.