matrices word problem help?
April 28, 2011
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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April 28th, 2011 at 10:50 am
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
.
April 28th, 2011 at 10:50 am
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 ]