Twenty meters of wire is available for fencing oft a flower
bed in the form of a circular sector. Then the maximun
area (in sq. m ) of the flower-bed, is

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Twenty meters of wire is available for fencing oft a flower bed in the form of a circular sector. Then the maximun area (in sq. m ) of the flower-bed, is

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To determine the maximum area of the flower bed formed as a circular sector with 20 meters of wire, let's follow these steps:

1. **Define Variables:**
   - Let $$ r $$ be the radius of the sector.
   - Let $$ \theta $$ be the central angle in radians.

2. **Perimeter Constraint:**
   The total length of the wire is used as the perimeter of the sector, which includes the two radii and the arc length:
   $$
   2r + r\theta = 20 \quad \Rightarrow \quad r(2 + \theta) = 20 \quad \Rightarrow \quad r = \frac{20}{2 + \theta}
   $$

3. **Area of the Sector:**
   The area $$ A $$ of the sector is given by:
   $$
   A = \frac{1}{2} r^2 \theta
   $$
   Substituting $$ r $$ from the perimeter constraint:
   $$
   A = \frac{1}{2} \left(\frac{20}{2 + \theta}\right)^2 \theta = \frac{200\theta}{(2 + \theta)^2}
   $$

4. **Maximizing the Area:**
   To find the value of $$ \theta $$ that maximizes $$ A $$, we take the derivative of $$ A $$ with respect to $$ \theta $$ and set it to zero:
   $$
   A(\theta) = \frac{200\theta}{(2 + \theta)^2}
   $$
   $$
   A'(\theta) = \frac{200(2 + \theta)^2 - 400\theta(2 + \theta)}{(2 + \theta)^4} = \frac{200(2 - \theta)}{(2 + \theta)^3}
   $$
   Setting $$ A'(\theta) = 0 $$:
   $$
   200(2 - \theta) = 0 \quad \Rightarrow \quad \theta = 2 \text{ radians}
   $$

5. **Calculating the Maximum Area:**
   Substituting $$ \theta = 2 $$ back into the expression for $$ r $$:
   $$
   r = \frac{20}{2 + 2} = 5 \text{ meters}
   $$
   Thus, the maximum area is:
   $$
   A = \frac{1}{2} \times 5^2 \times 2 = 25 \text{ square meters}
   $$

**Answer:**  
25
The maximum area of the flower bed is 25 square meters.

Twenty meters of wire is available for fencing oft a flower
bed in the form of a circular sector. Then the maximun
area (in sq. m ) of the flower-bed, is

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