Question

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

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

Answer 1

Answer: (D)

Let length of sector is land radius of sector is $r$.
$\therefore l=\frac{2\pi r\theta }{360^{\circ }}$
Perimeter of sector $P=\frac{2\pi r\theta }{360^{\circ }}+2r$
$\Rightarrow P=\left(\frac{2\pi \theta }{360^{\circ }}+2\right)r$
$\Rightarrow r=\frac{P}{\left(\frac{2\pi \theta }{360^{\circ }}+2\right)}$
$\because A=\frac{\pi r^2\theta }{360^{\circ }}$
$A=\frac{\pi }{360^{\circ }}\left[\frac{P^2}{{\left(\frac{2\pi \theta }{360^{\circ }}+2\right)}^2}\right]\theta$
$A=\frac{\pi P^2}{360^{\circ }}\left[\frac{\theta }{{\left(\frac{2\pi \theta }{360^{\circ }}+2\right)}^2}\right]$
$\frac{dA}{d\theta }=\frac{\pi P^2}{360^{\circ }}\left[\frac{{\left(\frac{2\pi \theta }{360^{\circ }}+2\right)}^2-\theta \cdot 2\left(\frac{2\pi \theta }{360^{\circ }}+2\right)\frac{\pi }{360^{\circ }}}{{\left(\frac{2\pi \theta }{360^{\circ }}+2\right)}^4}\right]$
Put $\frac{dA}{d\theta }=0$, for maxima or minima
$\left(\frac{2\pi \theta }{360^{\circ }}+2\right)-\frac{4\theta \pi }{360^{\circ }}=0$
$\Rightarrow \frac{\pi \theta }{180^{\circ }}=2$
$\Rightarrow \theta =\frac{2\times 180^{\circ }}{\pi }$
$=2$ rad
Thus Area of sector will be maximum, if sectorial angle is of $2$ rad.