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matrices word problem help?

Suppose you are making mosaic tiles from 3 types of stained glass. You need 6 square feet of glass for the project and you want there to be as much iridescent glass as red and blue glass combined. The cost of a sheet of glass having an area of 0.75 square foot is .50 for iridescent, .50 for red, and .50 for blue. How many sheets of each type should you purchase if you plan to spend on the project?

I don’t understand this. Can anyone help?

Thank you.


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    2 Comments

    1. let x=blue ,y=red and z=iridescent in square feet
      then
      ‘You need 6 square feet of glass’ says
      x+y+z=6……………(1)
      ‘you want there to be as much iridescent glass as red and blue glass combined’ says
      x+y=z
      that is
      x+y-z=0……………..(2)
      ‘The cost of a sheet of glass having an area of 0.75 square foot is $6.50 for iridescent, $4.50 for red, and $5.50 for blue. How many sheets of each type should you purchase if you plan to spend $45’ says
      $4.5/0.75ft^2 =$(4.5/0.75) /ft^2 = $6 /ft^2 for red, and similarly for the others so using this logic
      (4.5/0.75)x + (5.5/0.75)y + (6.5/0.75)z = 45
      multiplying through by 7.5 gives
      45x + 55y + 65z = 337.5…………..(3)

      so the matrix can be made from these 3 equations but it is difficult for me to do in matrix notation on yahoo so
      subtracting (2) from (1) gives
      2z=6 so
      z=3
      substituting this gives
      x+y=3…………………(1*)
      45x+55y= 142.5…(2*)
      then (2*) – 45*(1*) gives
      10y=7.5 so
      y=0.75
      sub y in (1*) gives
      x=3 – 0.75 = 2.25
      therefore
      x=2.25, y=0.75, z=3 all in ft^2
      so
      dividing x,y,z by 0.75 gives the number of sheets

      red=3, blue=1, iridescent=4

      the end
      .

    2. Ok, the information we can drag out of the words is that:
      a) the total area is 6 sq ft, so 0.75i + 0.75r + 0.75b = 6
      b) the area of iridescent = area blue + area red; so 0.75i=0.75r+0.75b or i – a – b = 0
      c) the total cost is (at most) $45; so 6.5i + 4.5r + 5.5b = 45

      you can then plug these three formulae into a matrix and use row reduction to determine the answer
      [0.75 0.75 0.75 6 ]
      [1 -1 -1 0 ]
      [6.5 4.5 5.5 45 ]

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