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.
To solve this problem, let's assign some variables:
Let x be the number of cubic yards of mix A.
Let y be the number of cubic yards of mix B.
We need to find the values of x and y that satisfy the given conditions.
First, let's establish the equations for the nutrient requirements:
For nitrogen:
Mix A contains 10 lbs of nitrogen per cubic yard, and mix B contains 5 lbs of nitrogen per cubic yard. So the equation for the nitrogen requirement becomes:
10x + 5y ≥ 350
For phosphoric acid:
Mix A contains 8 lbs of phosphoric acid per cubic yard, and mix B contains 24 lbs of phosphoric acid per cubic yard. So the equation for the phosphoric acid requirement becomes:
8x + 24y ≥ 630
For potash:
Mix A contains 9 lbs of potash per cubic yard, and mix B contains 6 lbs of potash per cubic yard. So the equation for the potash requirement becomes:
9x + 6y ≤ 840
Now, let's consider the cost:
A cubic yard of mix A costs $7, and a cubic yard of mix B costs $9. The total cost equation becomes:
7x + 9y = Total Cost
Since we want to find the minimum cost, this equation doesn't help us much. However, we can use it to determine the region bounded by the inequality equations.
Now that we have all the necessary equations, we can graph them on a coordinate plane.
1. Graph the equation 10x + 5y ≥ 350:
- Rewrite the equation as 5y ≥ -10x + 350.
- Plot the line -10x + 350 = 0.
- Shade the region above the line.
2. Graph the equation 8x + 24y ≥ 630:
- Rewrite the equation as 4x + 12y ≥ 315.
- Plot the line 4x + 12y = 315.
- Shade the region above the line.
3. Graph the equation 9x + 6y ≤ 840:
- Plot the line 9x + 6y = 840.
- Shade the region below the line.
4. Determine the region where all the shaded regions intersect. This intersection represents the valid solutions for the given constraints.
Finally, to find the optimal solution, you can substitute the corner points of the shaded region into the equation 7x + 9y = Total Cost and compare the costs. The solution with the minimum cost will give you the number of cubic yards of each mix to add to the field.
Note: I understand that graphing can be challenging to explain without visuals. It might be helpful to refer to your textbook or online sources that provide examples and illustrations of graphing inequalities in two variables.