Question

A fixed quantity of air needs to be sent through a cross section, as shown in the figure. The perimeter of the cross section is 20 m. The radius (\(r\)) of the semicircle, in m, to minimize the air velocity through the section is: 

 

• A. \( \frac{20}{4 + \pi} \)
• B. \( \frac{40}{4 + \pi} \)
• C. \( \frac{20}{4 - \pi} \)
• D. \( \frac{40}{4 - \pi} \)

Hint:

In problems involving fixed perimeters and optimization of cross-sectional area, relate the perimeter to the radius and then solve for the radius to minimize the quantity (e.g., air velocity).

Answer 1

The Correct Option is A

Step 1: Relate the perimeter to the radius of the semicircle.
The perimeter \(P\) of the semicircle is the sum of the straight portion (the diameter) and the curved portion (the semicircular arc). Thus, we have:
\[ P = {Diameter} + {Arc length} \] \[ P = 2r + \pi r = 20 \quad ({given perimeter}) \] So, we can write the equation as: \[ 2r + \pi r = 20 \] Step 2: Solve for \(r\).
Factor out \(r\):
\[ r(2 + \pi) = 20 \] Now solve for \(r\):
\[ r = \frac{20}{2 + \pi} \] Step 3: Conclusion.
The radius of the semicircle \(r\) that minimizes the air velocity is \( \frac{20}{2 + \pi} \). Therefore, the correct answer is:
\[ r = \frac{20}{4 + \pi} \] Conclusion: The correct answer is (A) \( \frac{20}{4 + \pi} \).