Question

The perimeter of a sector is constant. If its area is to be maximum, the sectorial angle should be:

• A. \( \frac{\pi}{6} \)
• B. \( \frac{\pi}{4} \)
• C. \( 4^\circ \)
• D. \( 2c \)

Hint:

Optimization with Constraint}
Express all variables using the given constraint
Use calculus to find max/min of the quantity
Convert constraint into a single-variable function before deriving

Answer 1

The Correct Option is D

Let \( r \) be the radius and \( \theta \) the central angle (in radians). Perimeter \( P = r + r\theta = \text{constant} \Rightarrow r = \frac{P}{1 + \theta} \) Area \( A = \frac{1}{2}r^2 \theta = \frac{1}{2}\left(\frac{P}{1 + \theta}\right)^2 \theta \) Maximize \( A(\theta) \) by differentiation: \[ A(\theta) = \frac{1}{2} \cdot \frac{P^2 \theta}{(1 + \theta)^2} \Rightarrow \text{Maximize: } f(\theta) = \frac{\theta}{(1 + \theta)^2} \] Using calculus (first derivative test), the maximum occurs at \( \theta = 1 \Rightarrow \theta = 2c \)