The perimeter of a sector is p. The area of the sector is maximum when its radius is

Options

A. `sqrtp`
B. `(1)/(sqrtp)`
C. `(p)/(2)`
D. `(p)/(4)`

Answer

D

Explanation

Perimeter of a sector = p. Let aOB be the sector with radius r.
If angle of the sector be `theta` radians, then area of sector
`(A)=(1)/(2)r^(2)theta" (i)"`
Length of arc(s) `=r theta or theta=(s)/(r).`
Therefore perimeter of the sector
`p=r+s+r=2r+s" (ii)"`
Substituting `theta=(s)/(r)` in (i) `A=((1)/(2)r^(2))((s)/(r))=(1)/(2)rs`
`p=2r+((2A)/(r))or 2A=pr-2r^(2)`.
Differentiating with respect to r, we get
`2(dA)/(dr)=p-4r`
For maximum value of area
`(dA)/(dr)=0 or p-4r = 0 or r=(p)/(4)`