Question

A circular sector of perimeter $60$ metre with maximum area is to be constructed. The radius of the circular arc in metre must be

• A. 10
• B. 15
• C. 5
• D. 20

Answer 1

Answer: (B)

Perimeter of sector = $2r + r \theta$
$\Rightarrow \, 60 = 2r + r\theta$ (given)

$\Rightarrow \; \theta = \frac{60 - 2r}{r}$
Area of sector, $A = \frac{\pi r^2 \theta}{360^{\circ}}$
$= \frac{\pi r^{2} \left(60 -2r\right)}{r 360} $
$ = \frac{\pi r}{180} \left(30 -r\right) $
$\Rightarrow \frac{dA}{dr} = \frac{\pi}{180} \left(30 -2r\right) $
For maximum area, $ \frac{dA}{dr} = 0 $
$ \Rightarrow 30-2r=0 $
$ \Rightarrow r = 15 $
$\therefore \frac{d^{2}A}{dr^{2}} = \frac{\pi}{180} \left(0-2\right) = \frac{-\pi}{90} <\, 0 $
$\therefore $ It is maximum at r = 15 m.