Question

Use the following information. The power generated by a windmill can be modeled by the equation \(w=0.015 s^{3},\) where \(w\) is the power measured in watts and \(s\) is the wind speed in miles per hour. Write a general statement about how doubling the wind speed affects the amount of power generated by a windmill.

Short answer

Doubling the wind speed will increase the power generated by the windmill by a factor of 8.

Step-by-step solution

Understand the Relationship

In the equation \(w=0.015 s^{3}\), \(w\) varies as the cube of \(s\). It means that if \(s\) is doubled, \(w\) will be increased by the cube of 2, which equals 8.

Calculate Power with Original Wind Speed

Let's denote the original wind speed as \(s\). The current power generated by the windmill is \(w_1 = 0.015 s^{3}\).

Calculate Power with Doubled Wind Speed

If the wind speed is doubled, the new wind speed is \(2s\). The power generated by the windmill with doubled wind speed is \(w_2 = 0.015 (2s)^{3}\). Simplifying yields \(w_2 = 0.015 * 8s^{3} = 8w_1\). Here, \(8w_1\) represents eight times the power generated with the original wind speed.

Key concepts

Wind Power Generation Equation

Understanding the wind power generation equation is essential for grasping how wind energy can be harnessed and converted into electricity. In the given exercise, the power, denoted by variable \(w\), is a function of wind speed \(s\). The relationship is expressed as \(w = 0.015 s^3\), where \(w\) represents power in watts and \(s\) is the wind speed measured in miles per hour (mph).

This equation is a simplified model that encapsulates the cubical relationship between the wind speed and the power output of a windmill. It's important to note that the coefficient, 0.015, can vary based on the design and efficiency of the wind turbine. It reflects how much power is produced for a given wind speed, taking into account factors like blade length, generator efficiency, and air density.

By analyzing this equation, one can predict how much electricity a windmill can generate under varying wind conditions, making it a crucial component in the planning and operation of wind power facilities.

Effects of Wind Speed on Power

Wind speed plays a pivotal role in the amount of power generated by a windmill. As evidenced by the provided exercise, if you were to double the wind speed, there would be a profound impact on the energy output. This is due to the cubic relationship between wind speed and power output described in the wind power generation equation.

To illustrate this, consider an original wind speed \(s\); the corresponding power output will be \(w_1 = 0.015 s^3\). On doubling the wind speed to \(2s\), the new power output becomes \(w_2 = 0.015 (2s)^3 = 0.015 * 8s^3 = 8w_1\). This significant increase is because power rises by the cube of any increase in wind speed; in this case, doubling wind speed raises the power by a factor of eight (since \(2^3 = 8\)).

This demonstrates a highly sensitive and nonlinear relationship, where even small increases in wind speed can lead to substantial gains in power generation. This sensitivity is a double-edged sword; it means wind turbines are highly efficient in harnessing wind at higher speeds but also susceptible to dramatic power drops when wind speeds fall.

Cubic Functions

Cubic functions are algebraic expressions of the form \(f(x) = ax^3 + bx^2 + cx + d\) where \(a\), \(b\), \(c\), and \(d\) are constants, and the highest power of \(x\) is three. The windmill power equation \(w = 0.015 s^3\) is a simplified cubic function with only the cubic term present.

Cubic functions are known for their distinctive S-shaped curves when plotted on a graph. They can have up to two turning points and exhibit varying behavior such as one x-intercept or three real x-intercepts. Cubic relationships indicate that for every unit increase in the independent variable (in the windmill equation, this is \(s\) or wind speed), the dependent variable (power output, \(w\)) increases by the cube of that unit.

The behavior of cubic functions like the one in our exercise highlights how outputs can change more dramatically than the inputs. This is much unlike linear functions, where the output changes in direct proportion to the input. In the context of wind power generation, this means small adjustments to wind speed can lead to large variations in power output, emphasizing the importance of optimal wind turbine placement and design for efficient energy production.