S2T2

Jim Randell 10:45 am on 25 August 2022 Permalink Reply
Tags: by: Des MacHale ( 11 )   

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  Jim Randell 10:46 am on 25 August 2022 Permalink | Reply

  Suppose the floors of the rooms have sides a, b, c (smallest to largest).

  Then, if the carpet from the largest bedroom can be used to exactly carpet the other 2 bedrooms we have:

  c² = a² + b²

  So, (a, b, c) form a right-angled triangle (but this is not necessarily a Pythagorean triple, as we are not told that the sides of the rooms take on integer values).

  Or:

  b² = c² − a²
  ⇒ b² = (c − a)(c + a)

  Each side of the larger square must be reduced (otherwise we have a rectangle with side c that won’t fit in the smaller bedrooms).

  Assuming the carpet is cut into exactly 4 rectangular regions (which may be square), we must have cuts like this:

  

  (i.e. one cut must go across the entire length of the square, and the rectangles produced from this cut must both have cuts perpendicular to this).

  I supposed we cut off an a × a square for the small room, which leaves three rectangles remaining to assemble into a square for the middle room.

  

  This gives us 3 remaining pieces of size (for some value d):

  a × (c − a)
  (c − a) × d
  (c − a) × (c − d)

  And in order to be assembled into a square of side b one of the pieces must have a dimension of b.

  So either: b = (c − a) or b = d.

  If b = (c − a) we infer that b = c, which is not possible, so the 3 remaining pieces are:

  a × (c − a)
  (c − a) × b
  (c − a) × (c − b)

  Which fit together like this:

  

  From which we see (vertical edges):

  b = a + (c − b)
  ⇒ 2b = a + c
  ⇒ b = (a + c) / 2

  i.e. b is the mean of a and c.

  So we write:

  b = a + x
  c = a + 2x

  To give the following diagram:

  

  From which we see (horizontal edges):

  a + x = 4x
  ⇒ a = 3x

  So:

  a = 3x
  b = 4x
  c = 5x

  (So (a, b, c) is a Pythagorean triple, if we measure it in units of x).

  And the pieces and their areas are:

  a × a = 9x²
  a × (c − a) = 6x²
  b × (c − a) = 8x²
  (c − b) × (c − a) = 2x²

  And the smallest of these (2x²) has an area of 4 square metres hence x² = 2.

  And we want the area of the largest bedroom (c²).

  c² = (5x)²
  ⇒ c² = 25x²
  ⇒ c² = 50 square metres

  Solution: The floor area of the largest bedroom is 50 square metres.

  ---

  And we can calculate the floor areas of the other rooms as well:

  small = 18 square metres (a 3√2 metre square)
  middle = 32 square metres (a 4√2 metre square)
  large = 50 square metres (a 5√2 metre square)

  And 18 + 32 = 50.

  The 50 square metre carpet from the largest room is cut into the following rectangles:

  √2 × 2√2 (≈ 1.41 × 2.83) = 4 square metres (b.1)
  2√2 × 3√2 (≈ 2.83 × 4.24) = 12 square metres (b.2)
  2√2 × 4√2 (≈ 2.83 × 5.66) = 16 square metres (b.3)
  3√2 × 3√2 (≈ 4.24 × 4.24) = 18 square metres (a)

  Or with each square having an area of 2 square metres (i.e. of side √2 metres):

  

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