Problem 33 - Algebra and Trigonometry

Question

Power from a Windmill 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) \( P = k s^3 \). (b) \( k = 0.012 \). (c) 324 watts.

Step 1: Understand the Proportionality Relationship

The power \( P \) from a windmill is directly proportional to the cube of the wind speed \( s \). This means we can express \( P \) in the form \( P = k s^3 \), where \( k \) is the constant of proportionality. This equation shows that any change in \( s \) will cause \( P \) to change by the cube of that factor, multiplied by \( k \).

Step 2: Find the Constant of Proportionality

We are given that when \( s = 20 \) mi/h, the power \( P \) is 96 watts. Using the equation \( P = k s^3 \), we substitute the known values: \( 96 = k \times 20^3 \). Calculating \( 20^3 \) gives \( 8000 \), so the equation becomes \( 96 = 8000k \). Solving for \( k \), we get \( k = \frac{96}{8000} = 0.012 \).

Step 3: Calculate Power at a New Wind Speed

With \( k = 0.012 \) determined, we now need to find the power when the wind speed \( s = 30 \) mi/h. Using the equation \( P = k s^3 \), substitute \( s = 30 \) and \( k = 0.012 \). Calculate \( 30^3 = 27000 \), so \( P = 0.012 \times 27000 = 324 \).

Step 4: Conclusion

The windmill will produce 324 watts of power when the wind speed is 30 mi/h, using the established proportionality constant \( k = 0.012 \) and the formula \( P = k s^3 \).

Key Concepts

Cube of Wind Speed

One integral concept to grasp in this problem is how wind speed affects power output through a cubed relationship. This means the power generated by the windmill doesn't just double if the wind speed doubles, nor does it triple if the wind speed triples. Instead, the power output increases as the cube of the wind speed.

For example:

• If the wind speed doubles, the power increases by eight times (since \(2^3 = 8\)).
• If the wind speed triples, the power increases by twenty-seven times (since \(3^3 = 27\)).

This concept is rooted in physics because the energy in wind is spread over a larger area and moves with more force as speed increases.

Therefore, when solving wind power problems, ensure that you are cubing the wind speed correctly. This exponential growth reflects how increasing wind speed significantly impacts power production.

Constant of Proportionality

In the context of this problem, direct proportionality indicates a consistent relationship between two variables. Here, the power from a windmill \( P \) is directly proportional to the cube of the wind speed \( s^3 \).

The constant of proportionality \( k \) acts as a multiplier that defines this specific relationship. Using the formula \( P = k s^3 \), you can find \( k \) by substituting known values for \( P \) and \( s \).

Given:

• Power \( P = 96 \) watts
• Wind speed \( s = 20 \) mi/h

Firstly, calculate \( s^3 \):
\( 20^3 = 8000 \).
Now you can solve for \( k \):
\( k = \frac{96}{8000} \equiv 0.012 \).
This constant helps predict how varying wind speeds will affect the power output of the windmill, maintaining a constant proportional relationship.

Power Calculation

Calculating the power output of a windmill involves understanding how to apply the formula \( P = k s^3 \) effectively. This equation uses both the constant of proportionality \( k \) and the wind speed \( s \), allowing you to compute the resulting power \( P \).

After determining \( k \) as \( 0.012 \), you can find the power output for different wind speeds. For instance, to calculate when the wind speed increases to \( 30 \) mi/h, we follow these steps:

• Cube the new wind speed: \( 30^3 = 27000 \).
• Use the formula with \( k = 0.012 \): \( P = 0.012 \times 27000 \).
• The result is \( P = 324 \) watts.

This demonstrates how efficiently the power can be calculated when both the relationship and the constant of proportionality are well understood.

Experimenting with different wind speeds allows you to appreciate the broad spectrum of power a windmill can generate, reinforced by consistent application of the formula.