Problem 95 - Algebra and Trigonometry

Question

Power Produced by a Windmill When the wind blows with speed \(v \mathrm{km} / \mathrm{h}\) , a windmill with blade length 150 \(\mathrm{cm}\) generates \(P\) watts \((\mathrm{W})\) of power according to the formula \(P=15.6 v^{3} .\) (a) How fast would the wind have to blow to generate \(10,000 \mathrm{W}\) of power? (b) How fast would the wind have to blow to generate \(50,000 \mathrm{W}\) of power?

Step-by-Step Solution

Answer

To generate 10,000W, wind speed is approx. 8.65 km/h, and for 50,000W, it's approx. 14.70 km/h.

Step 1: Understand the Formula

We know from the problem statement that the power generated by the windmill is given by the formula \(P = 15.6 v^3\), where \(P\) is the power in watts and \(v\) is the wind speed in kilometers per hour. We need to find the wind speed \(v\) that produces a specified power \(P\).

Step 2: Rearrange the Formula for V

To find \(v\), we need to rearrange the formula to solve for \(v\). Start by isolating \(v^3\): \[v^3 = \frac{P}{15.6}\]. Then take the cube root to solve for \(v\): \[v = \sqrt[3]{\frac{P}{15.6}}\].

Step 3: Calculate Wind Speed for 10,000W

Substitute \(P = 10,000\) into the rearranged formula: \[v = \sqrt[3]{\frac{10,000}{15.6}}\]. Calculate \(v\): \(v \approx \sqrt[3]{641}\). Evaluate this to find \(v \approx 8.65\) km/h.

Step 4: Calculate Wind Speed for 50,000W

Repeat the same substitution process for \(P = 50,000\).\[v = \sqrt[3]{\frac{50,000}{15.6}}\]. Compute \(v\): \(v \approx \sqrt[3]{3205}\). Simplifying, we get \(v \approx 14.70\) km/h.

Key Concepts

Understanding the Power Formula

Windmills convert the kinetic energy of wind into electrical power. This conversion is given by a specific power formula. In this exercise, the power output, denoted as \(P\), that a windmill generates is calculated using the equation \(P = 15.6 v^3\). Here, \(P\) represents power in watts, and \(v\) is the wind speed in kilometers per hour.

The relationship between wind speed and power output is cubic. This means if the wind speed doubles, the power output increases by eight times (since \(2^3 = 8\)). The constant \(15.6\) is a factor that considers the windmill's efficiency and the physical characteristics of the windmill blades. It's essential to recognize that small changes in wind speed significantly affect the power produced.

Calculating Wind Speed

To determine how fast the wind must blow to generate a certain amount of power, we need to manipulate the given power formula. We're looking for wind speed \(v\) when a specified power \(P\) is known.

Start by isolating the wind speed variable. Begin with the equation \(P = 15.6 v^3\). Divide both sides of the equation by 15.6 to isolate \(v^3\), which gives:

• \(v^3 = \frac{P}{15.6}\)

The next step involves solving for \(v\) by taking the cube root of both sides:

• \(v = \sqrt[3]{\frac{P}{15.6}}\)

This rearranged equation allows for the calculation of wind speed based on a desired power output. Calculating wind speed involves substituting the specific power in place of \(P\) and then solving the resulting expression.

The Role of the Cube Root

Taking the cube root is an essential step in adjusting the formula to solve for wind speed. The cube root operation (\(\sqrt[3]{\text{...}}\)) is necessary when dealing with equations where a variable is raised to the power of three.

The notation \(\sqrt[3]{x}\) refers to the number that, when multiplied by itself twice, returns \(x\). For instance, the cube root of 27 is 3, as \(3 \times 3 \times 3 = 27\). In the context of our problem, the cube root helps determine the precise wind speed \(v\) for a given power \(P\), as seen in the rearranged equation: \(v = \sqrt[3]{\frac{P}{15.6}}\).

Recognizing the use of the cube root helps in understanding how to transition from power output to wind speed effectively. It ensures that the relationship between wind speed and power output is respected and calculated accurately.