question_answer

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

A)  20                                         

B)  5

C)  15                                         

D)  10

Correct Answer: C

Solution :

Perimeter of sector \[=2r+r\theta \] \[\Rightarrow \]               \[60=2r+r\theta \]          (given)                 \[\Rightarrow \]               \[\theta =\frac{60-2r}{r}\] Area of sector, \[A=\frac{\pi {{r}^{2}}\theta }{{{360}^{o}}}\]                                 \[=\frac{\pi {{r}^{2}}(60-2r)}{r\,\,360}\]                                 \[=\frac{\pi r}{180}(30-r)\] \[\Rightarrow \]               \[\frac{dA}{dr}=\frac{\pi }{180}(30-2r)\] For maximum area, \[\frac{dA}{dr}=0\] \[\Rightarrow \]               \[30-2r=0\] \[\Rightarrow \]               \[r=15\] \[\therefore \]  \[\frac{{{d}^{2}}A}{d{{r}^{2}}}=\frac{\pi }{180}(0-2)=\frac{-\pi }{90}<0\] \[\therefore \] It is maximum at  \[r=15\text{ }m\].