Problem 99
Question
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 \(P=15.6 \mathrm{v}^{3}\). (a) How fast would the wind have to blow to generate 10,000 W of power? (b) How fast would the wind have to blow to generate 50,000 W of power?
Step-by-Step Solution
Verified Answer
To generate 10,000 W, the wind speed is about 8.64 km/h. To generate 50,000 W, it is about 14.74 km/h.
1Step 1: Understanding the problem
We need to find the wind speed \( v \) such that the power generated \( P \) is equal to specific amounts, 10,000 W and 50,000 W. We will use the given formula \( P = 15.6v^3 \) to solve for \( v \) in each case.
2Step 2: Setting up the equation for part (a)
For part (a), set \( P = 10,000 \) in the formula \( P = 15.6v^3 \). This gives us \( 10,000 = 15.6v^3 \).
3Step 3: Solving for the wind speed in part (a)
Rearrange the equation \( 10,000 = 15.6v^3 \) to solve for \( v \). This gives \( v^3 = \frac{10,000}{15.6} \). Calculate \( v^3 \) and then take the cube root to find \( v \).
4Step 4: Calculating the wind speed for part (a)
First, divide: \( \frac{10,000}{15.6} \approx 641.03 \). Next, take the cube root: \( v = \sqrt[3]{641.03} \approx 8.64 \). The wind speed is approximately 8.64 km/h to generate 10,000 W.
5Step 5: Setting up the equation for part (b)
For part (b), set \( P = 50,000 \) in the formula \( P = 15.6v^3 \). This gives us \( 50,000 = 15.6v^3 \).
6Step 6: Solving for the wind speed in part (b)
Rearrange the equation \( 50,000 = 15.6v^3 \) to solve for \( v \). This gives \( v^3 = \frac{50,000}{15.6} \). Calculate \( v^3 \) and then take the cube root to find \( v \).
7Step 7: Calculating the wind speed for part (b)
First, divide: \( \frac{50,000}{15.6} \approx 3205.13 \). Next, take the cube root: \( v = \sqrt[3]{3205.13} \approx 14.74 \). The wind speed is approximately 14.74 km/h to generate 50,000 W.
Key Concepts
Wind speedPower calculationCube rootAlgebraic equations
Wind speed
Wind speed is a critical factor in determining the power generated by wind turbines. It refers to how fast the air is moving in a particular direction.
For wind turbines, faster wind speeds generally mean more power can be produced. The power of the wind is extracted through the turbine blades, and this depends on the cube of the wind speed.
This means that even a small increase in wind speed can lead to a substantial increase in power generation. For example, if the wind speed doubles, the power output may increase eightfold due to the exponential effect of the cube.
For wind turbines, faster wind speeds generally mean more power can be produced. The power of the wind is extracted through the turbine blades, and this depends on the cube of the wind speed.
This means that even a small increase in wind speed can lead to a substantial increase in power generation. For example, if the wind speed doubles, the power output may increase eightfold due to the exponential effect of the cube.
Power calculation
The formula provided for power generation is essential for accurate calculations. In this problem, we use the formula:
\[ P = 15.6v^3 \]
Here, \( P \) represents the power in watts, and \( v \) is the wind speed in kilometers per hour (km/h).
The constant 15.6 is a factor that relates how efficiently the wind turbine converts wind into power.
\[ P = 15.6v^3 \]
Here, \( P \) represents the power in watts, and \( v \) is the wind speed in kilometers per hour (km/h).
The constant 15.6 is a factor that relates how efficiently the wind turbine converts wind into power.
- When calculating power, substitute the value of \( v \) (the wind speed) into the equation.
- It's important to understand this relationship because it informs us how changes in wind speed affect energy production.
Cube root
To solve for the wind speed in terms of power, you'll often encounter the cube root operation. After isolating \( v^3 \) in the power formula, you calculate \( v \) by taking the cube root.
For example, if your equation yields \( v^3 = 641.03 \), then you find \( v \) using:
\[ v = \sqrt[3]{641.03} \]
The cube root essentially asks, "What number, multiplied by itself three times, equals 641.03?" Calculators often have a cube root function (usually denoted by \( \sqrt[3]{} \)), which makes this computation straightforward. This step is crucial for translating a known power output back into the required wind speed.
For example, if your equation yields \( v^3 = 641.03 \), then you find \( v \) using:
\[ v = \sqrt[3]{641.03} \]
The cube root essentially asks, "What number, multiplied by itself three times, equals 641.03?" Calculators often have a cube root function (usually denoted by \( \sqrt[3]{} \)), which makes this computation straightforward. This step is crucial for translating a known power output back into the required wind speed.
Algebraic equations
Algebraic equations are key to figuring out wind speed for a given power output. Start with the given power formula, and substitute in the target power value. You then solve it algebraically to find the wind speed.
For instance, to find the wind speed for 10,000 W, start with:\[ 10,000 = 15.6v^3 \]
Solve for \( v^3 \) by dividing both sides by 15.6:\[ v^3 = \frac{10,000}{15.6} \]
Once \( v^3 \) is isolated, find \( v \) using the cube root. This sequence of steps is central to solving the problem, ensuring you handle the relationships between variables correctly.
For instance, to find the wind speed for 10,000 W, start with:\[ 10,000 = 15.6v^3 \]
Solve for \( v^3 \) by dividing both sides by 15.6:\[ v^3 = \frac{10,000}{15.6} \]
Once \( v^3 \) is isolated, find \( v \) using the cube root. This sequence of steps is central to solving the problem, ensuring you handle the relationships between variables correctly.
- Remember, isolating the variable (here, \( v \)) is crucial in algebra.
- Each equation requires careful manipulation to maintain equality.
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