Minimize the total area

Replies

• 

  silverscorpion

  Let's say the cut is made at a distance 'a' inches from any end..
  
  Then, we can have the first piece of length 'a' be made into a circle and the other piece of length (16-a) be made into a square. Or, we can have the piece of length a be made into a square and the other into a circle.
  
  In the first case, we have the total area as
  

  A = Pi * a[sup]2[/sup] + (16-a)[sup]2[/sup]

  In the second case, it is,
  

  A = Pi * (16-a)[sup]2[/sup] + a[sup]2[/sup]

  Taking either equation and minimising that equation with respect to a, we have,
  the total area will be minimum when the cut is made at
  
  

  a = 16/(1 + Pi) = 3.863 inches

  from one end, or
  
  

  a = (16 * Pi)/(1 + Pi) = 12.137 inches

  from the other end..
• 

  Banashree Patra

  silverscorpion

  Let's say the cut is made at a distance 'a' inches from any end..
  
  Then, we can have the first piece of length 'a' be made into a circle and the other piece of length (16-a) be made into a square. Or, we can have the piece of length a be made into a square and the other into a circle.
  
  In the first case, we have the total area as
  
  
  In the second case, it is,
  
  
  Taking either equation and minimising that equation with respect to a, we have,
  the total area will be minimum when the cut is made at
  
  from one end, or
  
  from the other end..

  #-Link-Snipped-# you made some silly mistake
  In the first case 'a' is the circumference of the circle.So its radius(r) would be a/(2*pi) and each side(y) of the square would be (16-a)/4
  So the total area=pi*r*r+y*y
  
  Similarly for second case.
• 

  silverscorpion

  ^^ ahh.. Yes, what a silly mistake indeed 😁
  
  So, the total area comes to be
  

  A = a[sup]2[/sup]/4*Pi + (16-a)[sup]2[/sup]/16

  Minimizing this, the value of a seems to be 16*Pi/(4+Pi) or 7.04 inches from one end..
• 

  Prashanth_p@cchi

  I think it has to be exact half. 8 inches each.
  Total area will be (min) = 4inches (square) + approx 5inches (circle) = approx 9 inches (9.09inches).
  
  Increasing the radius and decreasing the side of a square or the increasing the side of a square and decreasing the radius will result in a value more than ever since squares are involved in the calculation.
  
  Just take a n example and check for both the cases. I believe that I'm right.
  
  Let me know if I'm missing something.
• 

  Ramani Aswath

  SS is right. The 7.04 has to be made into a circle ad 8.96 into a square.