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
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. $0.0052 \mathrm{Av}^{2}$
Solution
Solution Steps
Step 1: Understand the Relationship
The problem states that the force F varies jointly as the blade area A and the square of the wind velocity v. This means we can express the relationship as:
F=k⋅A⋅v2
where k is the constant of proportionality.
Step 2: Substitute Known Values to Find k
We are given that F=19.6lb, A=3.76ft2, and v=31.7ft/s. Substitute these values into the equation to solve for k:
19.6=k⋅3.76⋅(31.7)2
Step 3: Solve for k
First, calculate (31.7)2:
(31.7)2=1004.89
Now substitute back into the equation:
19.6=k⋅3.76⋅1004.89
Solve for k:
k=3.76⋅1004.8919.6
Calculate the denominator:
3.76⋅1004.89=3778.4064
Now calculate k:
k=3778.406419.6≈0.005187
Step 4: Write the Final Equation
Substitute k back into the original equation:
F=0.005187⋅A⋅v2
Final Answer
The equation relating F, A, and v is:
0.005187Av2
Thus, the correct answer is B.