# Find the dimensions of a rectangle with area 1000 m^{2} whose perimeter is as small as possible

Area is defined as total space occupied by a surface and perimeter is defined as total length of rectangle.

## Answer: The dimensions of a rectangle with area 1000 m^{2} whose perimeter is as small as possible is 10√10m and 10√10m

Area of rectangle is defined as **length x breadth** and Perimeter of rectangle is defined as **2 times sum of length and 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,

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

Hence,

A = xy

Given, A = 1000

1000 = xy

⇒ y = 1000/x -------------------- (1)

We know that,

**Perimeter of a rectangle = 2(Length + Breadth)**

Hence,

P = 2(x + y) ------------------- (2)

Substituting y=1000/x from equation (1) in equation (2) we get,

P(x) = 2(x+1000/x)

= 2x+2000/x

Computing the derivative of P(x)

P'(x) = 2-2000/x^2

Finding the critical points,

2-(2000/x^{2} ) = 0

=> x^{2} = 1000

=> x = 10√10

Subtituting value of x in equation (1)

We get,

y = 1000/ 10√10

y = 10√10