Problem 59 - Algebra and Trigonometry with Analytic Geometry

Question

The power \(P\) (in watts) generated by a windmill that has efficiency \(E\) is given by the formula \(P=0.31 E D^{2} V^{3}\), where \(D\) is the diameter (in feet) of the windmill blades and \(V\) is the wind velocity (in \(\mathrm{ft} / \mathrm{sec}\) ). Approximate the wind velocity necessary to generate 10,000 watts if \(E=42 \%\) and \(D=10\).

Step-by-Step Solution

Answer

The wind velocity required is approximately 9.23 ft/sec.

Step 1: Convert Efficiency

To use the formula \(P = 0.31 E D^{2} V^{3}\), we need to convert the efficiency \(E\) from a percentage to a decimal. Since \(E = 42\%\), we have \(E = 0.42\).

Step 2: Substitute Known Values

We are given \(P = 10,000\) watts, \(E = 0.42\), and \(D = 10\) feet. Substitute these into the formula:\[10,000 = 0.31 \times 0.42 \times 10^2 \times V^3\]

Step 3: Simplify the Equation

Calculate \(D^2\) and multiply all constants together:\[10^2 = 100\]\[0.31 \times 0.42 \times 100 = 13.02\]Thus, the equation becomes:\[10,000 = 13.02 \times V^3\]

Step 4: Solve for \(V^3\)

Divide both sides by 13.02 to isolate \(V^3\):\[V^3 = \frac{10,000}{13.02} \approx 768.1\]

Step 5: Solve for \(V\)

Take the cube root of both sides to find \(V\):\[V = \sqrt[3]{768.1} \approx 9.23\]

Key Concepts

Wind Energy Calculations

Wind energy is an efficient and sustainable method of generating power. Understanding how energy is calculated is key to designing windmills that capture maximum energy. The formula used to calculate wind power generated by a windmill is critical:

• \(P = 0.31 E D^2 V^3\)

Here, \(P\) represents the power in watts. \(E\) is the efficiency, and it indicates how well the windmill converts wind energy into electric power. \(D\) is the diameter of the windmill blades in feet, symbolizing the area capturing wind. Lastly, \(V\) is the wind velocity measured in feet per second.
One main factor is that power output varies with the cube of the wind velocity \(V^3\). This means a slight change in wind speed results in a significant change in power generated. The larger the diameter \(D\) of the blades, the more wind is captured. Understanding these relationships allows us to grasp how different variables affect the performance of wind energy systems.

Efficiency Conversion

When dealing with formulas that include efficiency, it's crucial to express efficiency as a decimal rather than a percentage. If efficiency is given as a percentage, like 42%, you would convert it by dividing by 100, yielding 0.42.

• This step ensures that all variables in the equation are consistent

By converting the efficiency correctly, the calculations can accurately determine the output power.
Efficiency represents how effectively the windmill converts wind energy into electrical energy. Higher efficiency values indicate better conversion, meaning more of the wind's energy is turned into electricity rather than lost as heat or other forms of energy.
Thus, understanding and converting efficiency correctly is essential for solving wind energy problems and for optimizing windmill designs.

Cube Root Calculation

Sometimes, solving algebraic equations involves finding cube roots. In situations like wind energy calculations, this arises when isolating wind velocity. Once you have simplified an equation to look like \(V^3 = 768.1\), finding \(V\) involves calculating the cube root of 768.1.

• Cube root calculations help find one of the three numbers multiplied together to yield a given number

For example, \(V = \sqrt[3]{768.1} \approx 9.23\).
One strategy to approximate cube roots entails dividing the number into smaller, manageable parts and estimating through common cube values. However, using calculators or software tools provides quick and precise results.
Grasping cube roots helps in understanding the relationship between wind velocity and power output, allowing you to solve real-world problems like designing efficient wind turbines.