Question

Twenty metres of wire is available for fencing off a flower-bed in the form of a circular sector. Then the maximum area (in `s qdotm)` of the flower-bed is: 25 (2) 30 (3) 12.5 (4) 10

Answer 1

Let `r` is the radius of the circular sector and `theta` is the angle cretaed by it.
Here, perimeter of the circular sector `= r+r+rtheta = 2r+rtheta`
`:. 2r+rtheta = 20`
`=> theta = (20-2r)/r`
Now, area of the circlular sector,`A = 1/2r^2theta`
`A = 1/2r^2((20-2r)/r)`
`=>A = r(10-r) = 10r-r^2`
`=>(dA)/(dr) = 10-2r`
For maximum area, `(dA)/(dr)` should be `0`.
`:. 10 - 2r = 0 => r = 5m`
So, maximum area, `A_(max) = 10r - r^2 = 10(5) - (5)^2 = 25 m^2`