Problem 33 - Precalculus Mathematics for Calculus

Question

The power \(P\) that can be obtained from a windmill is directly proportional to the cube of the wind speed \(s\) (a) Write an equation that expresses this variation. (b) Find the constant of proportionality for a windmill that produces 96 watts of power when the wind is blowing at \(20 \mathrm{mi} / \mathrm{h}\) (c) How much power will this windmill produce if the wind speed increases to \(30 \mathrm{mi} / \mathrm{h}\) ?

Step-by-Step Solution

Answer

(a) \( P = k \cdot s^3 \); (b) \( k = 0.012 \); (c) 324 watts.

Step 1: Understanding the Proportionality

Since the power obtained from the windmill is directly proportional to the cube of the wind speed, we can express this relationship with the equation: \( P = k \cdot s^3 \), where \( k \) is the constant of proportionality and \( s \) is the wind speed.

Step 2: Solving for the Constant of Proportionality

Given that \( P = 96 \) watts when \( s = 20 \) mi/h, substitute these values into the equation: \( 96 = k \cdot 20^3 \). Simplify \( 20^3 \) to get \( 8000 \). Thus, \( 96 = k \cdot 8000 \). Solve for \( k \) by dividing both sides by 8000: \( k = \frac{96}{8000} = 0.012 \).

Step 3: Calculating Power at New Wind Speed

Now that we have \( k = 0.012 \), we find the power when the wind speed \( s = 30 \) mi/h. Substitute into the equation \( P = k \cdot s^3 \): \( P = 0.012 \cdot 30^3 \). Calculate \( 30^3 \) to get \( 27000 \). Thus, \( P = 0.012 \cdot 27000 = 324 \) watts.

Key Concepts

Power and Wind Speed Relationship

Power generation from a windmill relies significantly on the speed of the wind driving it. When we talk about the relationship between these two, we use the term "direct variation" to describe how one factor (power) changes consistently with changes to another factor (wind speed). Specifically, the power generated by a windmill is directly proportional to the cube of the wind speed.
This means that as the wind speed increases, the power output doesn't just increase linearly – it increases rapidly, as the cube of the speed. The mathematical expression of this relationship is given by the equation:
\[ P = k \cdot s^3 \]
Here, \( P \) represents the power, \( s \) is the wind speed, and \( k \) is the constant of proportionality. Simply put, for small increases in wind speed, there is a much larger increase in power due to the cubing effect, which highlights why wind speed is a crucial factor in wind energy production.

Proportionality Constant

The proportionality constant, denoted as \( k \), is an essential part of the equation that describes the direct variation between power and the cube of wind speed. It essentially quantifies the specific relationship between these variables for a given windmill or setup. To find this constant, we use known values of power and wind speed.
For example, if a windmill produces 96 watts of power at 20 mi/h wind speed, we can substitute these values into our equation:
\[ 96 = k \cdot 20^3 \]
Calculating \( 20^3 \) gives 8000, leading to the equation \( 96 = k \cdot 8000 \). Solving for \( k \), we divide 96 by 8000, obtaining a value of \( k = 0.012 \). This constant tells us how efficiently the particular windmill converts wind energy into electrical energy at the specific conditions provided. It is unique to the windmill's characteristics and is vital for predicting future power outputs should the wind speed change.

Cube of Wind Speed

Understanding why power is proportional to the "cube of wind speed" is crucial in grasping how windmills operate effectively. The mathematical term "cube" refers to raising a number to the power of three – that is, multiplying the number by itself twice more. In formula terms, if \( s \) is the wind speed, \( s^3 \) represents the cube of wind speed.
When calculating how much power a windmill can generate if the wind speed increases, the process involves calculating this cubic relationship. For instance, if the wind speed increases from 20 mi/h to 30 mi/h, we calculate \( 30^3 \), resulting in 27000. Substituting this back into our power equation \( P = 0.012 \times 27000 \) yields a result of 324 watts of power. This exponential increase, compared to the linear increase in wind speed, helps us understand why even small changes in wind speed can significantly affect wind power generation.