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, $A=\frac{\pi r^2\theta }{360^{\circ }}$
$=\frac{\pi r^2\left(60-2r\right)}{r360}$
$=\frac{\pi r}{180}\left(30-r\right)$
$\Rightarrow \frac{dA}{dr}=\frac{\pi }{180}\left(30-2r\right)$
For maximum area, $\frac{dA}{dr}=0$
$\Rightarrow 30-2r=0$
$\Rightarrow r=15$
$\therefore \frac{d^{2}A}{d{r}^2}=\frac{\pi }{180}\left(0-2\right)=\frac{-\pi }{90}<0$
$\therefore$ It is maximum at r = 15 m.