Question

For fencing of flower bed with 100 cm long wire in the form of circular sector, the maximum area of the flower bed is :

• A. 1000 cm²
• B. 225 cm²
• C. 625 cm²
• D. 500 cm²

Answer 1

The Correct Option is C

The problem involves optimizing the area of a flower bed formed as a circular sector with a given perimeter. A circular sector with radius \( r \) and central angle \( \theta \) (in radians) has a perimeter of \( r\theta + 2r \) which is equal to 100 cm.

We aim to express everything in terms of \( r \) and \( \theta \) to find the maximum sector area:

1. The perimeter equation is \( r(\theta + 2) = 100 \).

2. Solve for \( \theta \): \( \theta = \frac{100}{r} - 2 \).

3. The area \( A \) of the sector is given by:

\( A = \frac{1}{2} r^2 \theta = \frac{1}{2} r^2 \left(\frac{100}{r} - 2\right) \). Simplifying,

\( A = \frac{1}{2}(100r - 2r^2) = 50r - r^2 \).

4. To find the maximum area, differentiate \( A \) with respect to \( r \) and set the result to zero:

\( \frac{dA}{dr} = 50 - 2r = 0 \).

5. Solving gives \( r = 25 \) cm.

6. Substitute \( r = 25 \) into the \( \theta \) equation:

\( \theta = \frac{100}{25} - 2 = 2 \) radians.

7. Calculate the maximum area:

\( A = 50 \times 25 - 25^2 = 1250 - 625 = 625 \text{ cm}^2 \).

Thus, the maximum area of the flower bed is \( \mathbf{625 \ cm^2} \).