Problem 80 - Algebra and Trigonometry

Question

The power produced by a wind turbine depends on the speed of the wind. If a windmill has blades 3 meters long, then the power P produced by the turbine is modeled by $$P(v)=14.1 v^{3}$$ where \(P\) is measured in watts \((\mathrm{W})\) and \(v\) is measured in meters per second \((\mathrm{m} / \mathrm{s}) .\) Graph the function \(P\) for wind speeds between 1 \(\mathrm{m} / \mathrm{s}\) and 10 \(\mathrm{m} / \mathrm{s}\).

Step-by-Step Solution

Answer

Plot the points \( P(v) \) from \( v = 1 \) to \( v = 10 \) using \( P(v) = 14.1 v^3 \) and draw the curve.

Step 1: Understand the Function

The power function is given as \( P(v) = 14.1 v^3 \). This function relates the wind speed \( v \) to the power \( P \) generated by the wind turbine. We need to plot \( P \) for wind speeds between \( 1 \, \mathrm{m/s} \) and \( 10 \, \mathrm{m/s} \).

Step 2: Determine the Range of Values

Identify the range of \( v \) values to calculate \( P(v) \). In this case, you need to find \( P(v) \) for each integer value of \( v \) from \( 1 \, \mathrm{m/s} \) to \( 10 \, \mathrm{m/s} \).

Step 3: Calculate Power Values

For each wind speed \( v \) from 1 to 10, calculate \( P(v) \) using the formula. For example, when \( v = 1 \), \( P(1) = 14.1 \times 1^3 = 14.1 \) watts. Similarly, calculate \( P(2), P(3), \dots, P(10) \).

Step 4: Generate the Data Points

Create a list of data points \((v, P(v))\) that you will plot on the graph. Each data point corresponds to a calculated \( P(v) \) from the previous step.

Step 5: Plot the Graph

Using graphing software or graph paper, plot the points \((v, P(v))\) to visualize the function. Draw a smooth curve through the points to represent the function \( P(v) = 14.1 v^3 \). The graph should show exponential growth as \( v \) increases from 1 to 10.

Key Concepts

Wind Power Calculation

Wind power calculation is an important concept in understanding how wind energy is converted into electrical energy by a wind turbine. The power generated by a turbine is not a linear relationship but is instead expressed through a cubic function. Specifically, this means the power output increases dramatically as wind speed increases. This is because the power is proportional to the cube of the wind speed. Using the formula given in the problem, \( P(v) = 14.1 v^3 \), where \( P \) represents power measured in watts and \( v \) represents wind speed measured in meters per second, one can calculate the power output for various wind speeds.

Here’s how you can approach it:
- For each wind speed, substitute the value into the equation to get the power output.
- Due to the cubic relationship, small increases in wind speed can lead to significantly larger increases in power generated.

This is why precision in wind speed measurement is crucial for accurately predicting wind turbine power outputs. This non-linear relationship highlights the efficiency gains possible with even modest improvements in wind speed.

Wind Speed

Wind speed is a critical factor in determining the potential energy that can be harvested by a wind turbine. The formula \( P(v) = 14.1 v^3 \) demonstrates that power output is incredibly sensitive to changes in wind speed. For instance, doubling the wind speed from 2 m/s to 4 m/s results in an eightfold increase in power. This is crucial because turbines must be carefully positioned in areas with optimal wind conditions to maximize efficiency and output.

Understanding wind speed involves:
- Measuring accurately using anemometers.
- Analyzing historical wind data to predict energy yield.

Even small geographic differences can lead to significant variance in available wind resources, impacting decision-making on turbine placement and the economic feasibility of a project. Hence, accurate modeling of wind speed is crucial for wind farms to operate effectively.

Mathematical Modeling

Mathematical modeling plays a vital role in predicting and understanding the power generated by wind turbines. By using the formula \( P(v) = 14.1 v^3 \), we can simulate how different wind speeds will impact power output. This mathematical function is a cubic graph, which starts with a gentle curve and increases steeply as wind speed grows.

Key aspects of mathematical modeling in wind energy include:
- Using data to project potential energy outputs.
- Applying cubic functions to reflect the relationship between speed and power.

Graphing these functions helps visualize performance, making it easier to predict how minor changes in input (wind speed) can lead to major changes in output (power). Mathematical modeling is a powerful visualization tool, allowing engineers and planners to optimize the design and placement of turbines to harness wind energy more effectively. Additionally, it aids in forecasting and decision-making, ensuring resources are allocated efficiently.