Question 1075426
i'll provide the solution using the calculator at www.desmos.com.
this calculator allows you to graph inequalities and also allows you to present the equations as ax + by = c rather than y = (c-ax)/b.


the general procedure is:


determine the constraint functions.


graph the opposite of the inequalities.


the feasible region will be the area of the graph that is NOT shaded.


the corner points of the feasible region will contain the max / min solution.


you evaluate the objective function at those corner points and then choose the corner point that gives you the max/min solution.


you then check the constraint to see that they are all satisfied at that corner point.


you will see what i mean as we go through the problem.


your objective function is profit = 2x + 4y.


your constraint functions are:


x >= 0
y >= 0
.2x + .25y <= 240
x + y >= 1000


x + y >= 1000 means the number of pizza have to be greter than or equal to 1000.


.2x + .25y <= 240 means that the hours of labor to make the small pizza and the hours of labor to make the large pizza have to be less than or equal to 240.


x >= 0 and y >= 0 mean that the number of pizzas made can't be negative.


you will graph the opposite of the inequalities shown.


you will graph:


x <= 0
y <= 0
.2x + .25y >= 240
x + y <= 1000


each corner point is shown as (x,y), where x is the value of the x-coordinate of the point and y is the value of the y-coordinate of the point.


the corner points are at (200,800), (1000,0) (1200,0)


the value of the objective function when x = 200 and y = 800 will be:


profit = 2*200 + 4*800 = 400 + 3200 = 3600.


this will be your maximum profit.


you can calculate the objective function at each of the other corner points to see that this is true.


the value of your constraint functions when x = 200 and when y = 800 will be:


.2x + .25y = .2*200 + .25*400 = 240
x = 200
y = 800
x + y = 1000.


all the constraints are met.
x and y are >= 0
.2x + .25y is <= 240.
x + y is >= 1000.


your graph will look like this:


the first graph is the far out view.
the second graph is the near end view showing you the corner points of the feasible region.


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