# Find the dimensions of a rectangle with perimeter 100 m whose area is as large as possible.

The area of a rectangle is defined as the total space occupied by it and perimeter is defined as the total length of the boundary of the rectangle.

## Answer: The dimensions of the rectangle with a perimeter of 100m, and as large an area as possible, are 25 m and** **25 m.

Area of rectangle = **length × breadth**

Perimeter of a rectangle = **2 (length + breadth)**

**Explanation:**

Let 'A' be area and 'P' be perimeter of the rectangle

Let 'x' be the width and 'y' be the length

We know that,

**Perimeter = 2 (length + breadth)**

Hence,

P = 2(x+y)

=> 100 = 2(x+y) (Since, Perimeter = 100m)

=> x + y = 50

=> y = 50 - x ----------------------------- (1)

We know that,

**Area of a rectangle = Length × Breadth**

Hence,

A = xy -------------------(2)

By substituting the value of y from equation (1) to equation (2) we get,

Area = x(50 - x )

Area A(x) = 50 x - x^{2}

Computing the derivative of A(x) we get,

A'(x) = 50 - 2x

Finding the critical points,

50 - 2x = 0

=> 2 x = 50

=> x = 25

Substitute x = 25 in equation (1)

We get,

y = 50 - x

=> y = 25

### So, the rectangle with maximum area is a square with side lengths 25m

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