Question

20 meters of wire is available to fence a flower bed in the form of a circular sector. If the flower bed should have the greatest possible surface area, then the radius of the circle is

• A. 2m
• B. 4m
• C. 5m
• D. 10m

Hint:

When maximizing the area of a circular sector, relate the length of the wire to the radius and angle, and optimize accordingly.

Answer 1

The Correct Option is C

Step 1: Formula for the area of a sector.
The area of a circular sector is given by: \[ A = \frac{1}{2} r^2 \theta \] where \( r \) is the radius and \( \theta \) is the central angle in radians. The circumference of the sector is \( 2r \sin \frac{\theta}{2} \). We are given that the total length of the wire is 20 meters, so we have: \[ 2r \sin \frac{\theta}{2} = 20 \] Step 2: Maximize the area.
Maximizing the area with respect to \( r \) and \( \theta \), we get \( r = 5 \).
Step 3: Conclusion.
The correct answer is (C) 5m.