What is the area of the largest rectangle field which can be enclosed with 200 m fencing ?

Options

A. 1600 `m^(2)`
B. 2100 `m^(2)`
C. 2400 `m^(2)`
D. 2500 `m^(2)`

Answer

D

Explanation

Area of largest rectangular field for a given perimeter for is possible if length and breadth of rectangular field are equal i.e it is a square
`rarr 4x=200 rarr x =(200)/(4) =50m`
`therefore` Area of largest rectangular field = `50xx50=2500 m^(2)`
Let length and breadth of rectangular field be x and y respectively
`therefore` 2(x+y)=200
`rarr y =100 -x`
and area A=xy
`=x(100-x)=100x-x^(2)`
`before (dA)/(dx)=100-2x`
Now `(d^(2)A)/(dx^(2))=-2lt0`
which shows maximum indepenent of values of x and y but only when they are equal
`therefore` A is maximum at x=50
Hence required area `=50 (100-50)=50xx50=2500m^(2)`