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The perimeter of a sector is p. The area...

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

A

`sqrtp`

B

`(1)/(sqrtp)`

C

`(p)/(2)`

D

`(p)/(4)`

Text Solution

Verified by Experts

The correct Answer is:
D

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)`
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