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

The Correct Option is 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}