Question

Show that out of all rectangles with given area a square has minimum perimeter.

Answer 1

Answer:

Hence rectangle is square

Step-by-step explanation:

Rectangle Sides L & B

Perimeter = 2(L + B)

Area = A  (constant)

A = LB

=> B = A/L

Perimeter = 2( L + A/L)

P = 2( L + A/L)

dP/dL = 2(1  - A/L²)

dP/dL = 0 => 2( 1 - A/L² )= 0  => L² = A/

=> L = √A

d²P/dL² =  + 2A/L³  is + Ve

so  L = √A will give minimum perimeter

L = √A

=> B = A/√A  = √A

=> L = B = √A

Hence rectangle is square