Questions: The force F on the blade of a wind generator varies jointly as the blade area A and the square of the wind velocity v. Find the equation relating F, A, and v if F=19.6 lb when A=3.76 ft^2 and v=31.7 ft / s. Choose the correct answer below. A. 0.00519 A v^2 B. 0.005187 Av^2 C. 0.005 Av^2 D. 0.0052 Av^2

The force F on the blade of a wind generator varies jointly as the blade area A and the square of the wind velocity v. Find the equation relating F, A, and v if F=19.6 lb when A=3.76 ft^2 and v=31.7 ft / s.

Choose the correct answer below.
A. 0.00519 A v^2
B. 0.005187 Av^2
C. 0.005 Av^2
D. 0.0052 Av^2
Transcript text: The force $F$ on the blade of a wind generator varies jointly as the blade area $A$ and the square of the wind velocity $v$. Find the equation relating $F, A$, and $v$ if $F=19.6 \mathrm{lb}$ when $A=3.76 \mathrm{ft}^{2}$ and $v=31.7 \mathrm{ft} / \mathrm{s}$. Choose the correct answer below. A. $0.00519 A v^{2}$ B. $0.005187 \mathrm{Av}^{2}$ C. $0.005 \mathrm{Av}^{2}$ D. $0.0052 \mathrm{Av}^{2}$
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Solution

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Solution Steps

Step 1: Understand the Relationship

The problem states that the force F F varies jointly as the blade area A A and the square of the wind velocity v v . This means we can express the relationship as: F=kAv2 F = k \cdot A \cdot v^2 where k k is the constant of proportionality.

Step 2: Substitute Known Values to Find k k

We are given that F=19.6lb F = 19.6 \, \text{lb} , A=3.76ft2 A = 3.76 \, \text{ft}^2 , and v=31.7ft/s v = 31.7 \, \text{ft/s} . Substitute these values into the equation to solve for k k : 19.6=k3.76(31.7)2 19.6 = k \cdot 3.76 \cdot (31.7)^2

Step 3: Solve for k k

First, calculate (31.7)2 (31.7)^2 : (31.7)2=1004.89 (31.7)^2 = 1004.89

Now substitute back into the equation: 19.6=k3.761004.89 19.6 = k \cdot 3.76 \cdot 1004.89

Solve for k k : k=19.63.761004.89 k = \frac{19.6}{3.76 \cdot 1004.89}

Calculate the denominator: 3.761004.89=3778.4064 3.76 \cdot 1004.89 = 3778.4064

Now calculate k k : k=19.63778.40640.005187 k = \frac{19.6}{3778.4064} \approx 0.005187

Step 4: Write the Final Equation

Substitute k k back into the original equation: F=0.005187Av2 F = 0.005187 \cdot A \cdot v^2

Final Answer

The equation relating F F , A A , and v v is: 0.005187Av2 \boxed{0.005187 \, A \, v^2} Thus, the correct answer is B.

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