Find a rectangle {x,y} of area A and of smallest perimeter among such rectangles.

Question

Let A be a positive number. A rectangle is defined by a pair of numbers {x, y} that represent the lengths of its sides. The perimeter of the rectangle is the value 2(x + y). Find a rectangle {x,y} of area A and of smallest perimeter among such rectangles. Explain and show your calculations.

Answer 1

Perimeter of the rectangle = 2(x +y)

Area of the rectangle(A) = x * y = > y = (A / x) - - - ->(1)

Perimeter (p) = 2(x +y)

p = 2(x + (A /x))

To find the critical point (dp /dx ) = 0.

p = 2x + (2A /x)

dp / dx = 2 + 2A (-1 / x²) 

=> dp  / dx = 2 -2A / x²  [ Since d/dx (1 /x) = -1 / x² ]

Now dp / dx =0

2 - (2A / x²) = 0.

2 = (2A / x²) = > A = x² 

=> - - - ->(2)

Since A should be a positive number we consider 

Substitute (2) in (1)

y = A /  

Applying conjugate the value of  y becomes  = > y = .

Perimeter of the rectangle = 2(x +y) = 2(√A + √A ) = 2*2√A = 4√A

Hence the rectangle has  area 'A'  and the smallest perimeter is   4 .