Question

A person has a rectangular sheet of metal. He has to make cylindrical vessel whose both ends are closed and circular. When he minimizes the wastage of sheet what is the ratio of wasted sheet to utilized sheet

Answer 1

ratio of wasted sheet to utilized sheet = L/(2πB - L)

Step-by-step explanation:

Let say  Dimension of Rectangular Sheet = L & B

Area of Sheer  = LB

Let say L is the circumference of circle at ends of  cylinder

the Diameter = L /π

Height of cylinder = B - L/π

Utilized area  = L * (B - L/π)   + 2* π(L/2π)²   =   LB  - L²/π   + L²/2π

=  LB - L²/2π

Wasted Area =   (L/π)(L)   -   2π(L/2π)²   =  L²/π  -  L²/2π

= L²/2π

or Wasted area = LB - (LB - L²/2π  )   = L²/2π

ratio of wasted sheet to utilized sheet = (L²/2π) / ( LB - L²/2π )

=  (L²) / ( 2πLB - L² )

= L/(2πB - L)