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Question

The maximum area of a rectangle (in m2) having perimeter 1000 m is

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Solution

Let the length of the rectangle be x m, width be y m and area be A m2
The perimeter of the rectangle is 1000 m,
2x+2y=1000
y=500x(1)
Area of rectangle, A=xy
Thus A=x(500x)=500xx2(2)
To get the stationary point at which A will be maximum, we will differentiate equation (2),
dAdx=5002x=0
x=250 m
Substituting the value of x in (1), we get
y=250 m
Hence maximum area, A=xy=250×250=62500 m2

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