How many of each do you make to use up all the resources?
How many of each do you make to use up all the resources?
Okay, a company makes three types of patio furniture: chairs, rockers, and chaise lounges, oh my!
Each requires wood, plastic, and aluminum in the amounts shown in the table below. The company has 400 units of wood, 600 units in plastic, and 1500 units of aluminum. Naturally, the company wants to use up all of this. How many chairs, rockers, and chaise lounges (whatever that is) should it make?
chai r= 1 unit of wood - 1 units of plastic - 2 units of aluminum
rocker = 1 unit of wood - 1 units of plastic - 3 units of aluminum
chaise lounge = 1 unit of wood - 2 units of plastic - 5 units of aluminum
Best answer:
Answer by xtchsbrlx
Ok, so you have 3 types of chairs, each with a certain resource cost.
So, if you take the number made of each chair and multiply that number by the amount of resources used, then you will get the total amount of resources in the total production of that line of furniture. If you add all 3 totals together, you will get a total amount that will equal 400 wood, 600 plastic, and 1500 aluminum. Make sense?
c = # chairs, r = # rocker, l = # chaise lounges (what the heck are these)
w = unit of wood, p = unit of plastic, a = unit of aluminum
c(1w+1p+2a) + r(1w+1p+3a) + l(1w+2p+5a) = 400w + 600p + 1500a
[simplify]
cw + cp + 2ca + rw + rp + 3ra + lw + 2lp + 5la = 400w + 600p + 1500a
[recombine so that the resources are the same and then factor out the "units of resources" variable]
(cw + rw + lw) + (cp + rp + 2lp) + (2ca + 3ra + 5la) = 400w + 600p + 1500a
w(c+r+l) + p(c+r+2l) + a(2c + 3r + 5l) = 400w + 600p + 1500a
Ok, I’m actually not sure how to explain this, but basically what you can do is break apart this big complex equation into individual parts by setting each variable equal to each other on both sides. So set the “w” coefficients equal to each other on both sides, then set the “p” coefficients equal to each other, etc. You can do this I think because I’ve isolated all the terms into this symmetrical form, but honestly I’m not sure. But I did the work and checked it and it works out.. so whatever. The other time I know you can do this is when you have like 5x^2 and stuff - those coefficients must be equal on both sides of the equation as well. But.. under certain circumstances… or maybe the math just works out regardless. I don’t remember, sigh >_<
[Anyway, so this is what happens, I split the equation into 3 parts and then I canceled out the a/w/p variables]
c+r+l = 400
c+r+2l = 600
2c+3r+5l = 1500
[subtract first from second]
l = 200
[substitute l into the 2nd and 3rd equations and then multiply the 2nd by 2, then subtract 2nd from 3rd]
2c + 3r + 5*200 = 1500
2c + 2r + 4*200 = 1200
r = 100
[plug l and r back into 1st equation]
c = 100
c = 100, r = 100, l = 200
check it, and it works
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