posted by Michael on .
Farmer Fred can use two types of plant fertalizers, mix A and mix B. Type A contains 10 lbs of nitrogen, 8 lbs of phosphoric acid, and 9 lbs of potah in a cubic yard of each mix. Type B contains 5 lbs of nitrogen, 24 lbs of phosphoric acid and 6 lbs of potash in a cubic yard of each mix. Tests performed on the soil in a large field indicates that the field needs no more than 840 lbs of potash. The tests also indicate that at least 630 lbs of phosphoric acid and at least 350 lbs of nitrogen should be added to the field. A cubic yard of mix A costs $7 and a cubic yard of mix B costs $9. How many cubi yards of each mix should Farmer Fred add to the field in order to supply the necessary nutrients at minimal costs?
I need to graph this as a linear inequality and get the necessary equations! I have NO IDEA how to do this, and it's homework that's due tomorrow! PLEASE HELP!
I would make the horizontal axis mix A, the vertical axis mix B.
Now you have three lines..
1) Nitrogen line. If you supply all with A, it will be at least 35cubicyards, if all mix B, it will be at least 70. So connect those points on A and B, you know you will be to the right of that line.
2) potash. If you use all mix B, no more than 140cubic yards, if you use all mix A, no more than 840/9 cubic yards. Connect those points. Note you have to be to the left of this line.
3) phosphoric acid. Do the same technique to get this line, you have to be to the right of that line.
Now look at the area enclosed by the lines (left of some, right of others).
There is a very famous theorem that says the minimal cost will be at one of the corners, so test the cost function at each corner (knowing how many yards of A,B) is at that point.