Question 1137721: A farmer can buy two types of plant food, mix A and mix B. Each cubic yard of mix A contains 52 pounds of phosphoric acid, 27 pounds of nitrogen, and 8 pounds of potash. Each cubic yard of mix B contains 13 pounds of phosphoric acid, 27 pounds of nitrogen, and 16 pounds of potash. The minimum monthly requirements are 520 pounds of phosphoric acid, 810 pounds of nitrogen, and 320 pounds of potash. Find the set of feasible solutions graphically for the amounts of mix A and mix B that can be used. If x is the number of cubic yards of mix A used and y is the number of cubic yards of mix B used, write a system of linear inequalities that indicates appropriate restraints on x and y. Write an inequality for the constraint on phosphoric acid
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! x = number of cubic yards of mix A.
y = number of cubic yards of mix B.
set up a table as follows:
mix A mix B requirement
number of cubic yards x y >= 0
pounds of phosphoric acid 52 13 >= 520
pounds of nitrogen 27 27 >= 810
pounds of potash 8 16 >= 320
your constraint equations are:
52x + 13y >= 520
27x + 27y >= 810
8x + 16y >= 320
x >= 0
y >= 0
using the decmos.com calculator, you wold graph the opposite of these inequalities.
the area of the graph that is not shaded is your region of feasibility.
the corner points of the feasibility region contain the maximum / minimum values for the objective function, if you have one.
the objective function coulde be maximize profit, minimize cost, or any other such business objective.
this problem doesn't have one, so we won't evaluate the objective function at each of the corner points.
we'll just graph the region of feasibility and identify the corner points.
here's what the graph looks like.
the region of feasibility appears to be open ended because there is no limit on the maximum amount of ingredients that can be used; there is only a limit on the minimum amount of ingredients that can be used.
the corner points of the feasible region are (0, 53.33...), (40,0).
the choices are therefore, either all of mix A or all of mix B.
there doesn't appear to be a choice of some of mix A and some of mix B, as is normally seen with other type problems.
if the objective function was to minimize cost, then you would have a cost per cubic yard of mix ?A and a cost per cubic yard of mix B.
you would then evaluate the objective function at each of the corner points and select the corner point that has the minimum cost.
for example, if the cost for a cubic yard of mix A was 100 dollars and the cost for a cubic yard of mix B was 50 dollars, you would see that (0,53.33) would cost 50 * 53.33 and (40,0) would cost 100 * 40 and you would pick mix A because the cost for mix A would 2667 and the cost for mix B would be 4000.
if the cost for each mix were 100, then you would pick mix B because mix A would cost 5333 and mix B would cost 4000.
in a minimize / maximize type problem, you identify the corner points and then evaluate the objective function at those corner points.
your objective in this problem is to write a system of linear inequalities that indicates appropriate restraints on x and y. Write an inequality for the constraint on phosphoric acid.
that was done above.
the system of linear inequalities is:
52x + 13y >= 520
27x + 27y >= 810
8x + 16y >= 320
x >= 0
y >= 0
appropriate restraints on x and y are that they both have to be greater than or equal to 0.
linear inequality for phosphorous is 52x + 13y >= 520
note that the method used with the desmos.com calculator is not available with a lot of other graphing software.
most other software doesn't allow you to graph inequalities.
most other software requires you to transform the equation into slope intercept form of y = mx + b.
using the desmos.com software makes the job of graphing inequalities easy and will definitely spoil you once you learn how to use it.
the technique of graphing the opposite of the inequalities is recommended because it's much easier to spot the feasibility region than graphing the inequalities as is.
with a lot of other software, you need to convert the inequalities to y = mx + b form and graph the equality portion of the inequalities and then manually shade the area of the graph that satisfies the inequalities.
there's a lot more manual work involved that is eliminated through the use of the desmos.com calculator.
if you cannot use the desmos.com calculator, or have to graph using some other software, then come back to me and i'll show you how to do it.
|
|
|
