Question 1157317: At Taco Town, a taco contains 2 oz of ground beef and 1 oz of chopped tomatoes. A burrito contains 1 oz of
ground beef and 3 oz of chopped tomatoes. Near closing time the cook discovers that they have only 22 oz of ground
beef and 36 oz of tomatoes left. Then manager directs the cook to use the available resources to maximize their
revenue for the remainder of the shift. If a taco sells for 40 cents and a burrito sells for 65 cents, then how many of
each should they make to maximize their revenue?
Found 2 solutions by ikleyn, greenestamps:
Answer by ikleyn(54018)   (Show Source):
You can put this solution on YOUR website! .
At Taco Town, a taco contains 2 oz of ground beef and 1 oz of chopped tomatoes.
A burrito contains 1 oz of ground beef and 3 oz of chopped tomatoes.
Near closing time, the cook discovers that they have only 22 oz of ground beef and 36 oz of tomatoes left.
The manager then directs the cook to use the available resources to maximize revenue for the remainder of the shift.
If a taco sells for 40 cents and a burrito sells for 65 cents,
then how many of each should they make to maximize their revenue?
~~~~~~~~~~~~~~~~~~~~~~~~~
Let x be the number of tacos,
y be the number of burritos.
Then we have this maximization problem:
Maximize the revenue z = 40x + 65y under these restrictions
2x + y <= 22, (the ground beef restriction)
x + 3y <= 36, (the chopped tomatoes restriction)
x >= 0, y >= 0, x and y are integer numbers.
To get the feasibility domain, draw the lines in the first quadrant (see the plot below)
2x + y = 22 (red line),
x + 3y = 36 (green line).
Obviously, the feasibility domain is the set of points inside the quadrilateral     in the first quarter
including all its boundary points.
The vertices of this quadrilateral are O = (0,0), = (0,12), = (6,10), = (11,0).
The coordinates are easily found as the intersections of the corresponding lines.
Finding these coordinates is traditionally considered as elementary operations for LP problems,
so I will not bore the reader with these details.
Next, according to the "corner points method", we should evaluate the objective function z = 40x + 65y
at the corner points of the feasibility domain and select that vertex where the objective function is maximal.
The values of the objective function at the corner points are
z(O) = 40*0 + 65*0 = 0,
z(P1) = 40*0 + 65*12 = 780,
z(P2) = 40*6 + 65*10 = 890,
z(P3) = 40*12 + 65*0 = 480.
Thus the maximum value of the objective function at the corners is 890.
So, x = 6, y = 10 is the solution to this maximization problem in integer numbers,
giving the maximum value of 890 cents, or $8.90, for the revenue.
ANSWER. x = 6 (six tacos), y = 10 (ten burritos) is the solution, giving the maximum revenue of $8.90.
At this point, the solution is complete.
Answer by greenestamps(13382)  (Show Source):
You can put this solution on YOUR website!
Again for the beginning of my response I borrow from the solution from tutor @ikleyn:
Let x be the number of tacos,
y be the number of burritos.
Then we have this maximization problem:
Maximize the revenue z = 40x + 65y under these restrictions
2x + y <= 22, (the ground beef restriction)
x + 3y <= 36, (the chopped tomatoes restriction)
x >= 0, y >= 0, x and y are integer numbers.
To get the feasibility domain, draw the lines in the first quadrant (see the plot below)
2x + y = 22 (red line),
x + 3y = 36 (green line).
Obviously, the feasibility domain is the set of points inside the quadrilateral in the first quarter
including all its boundary points.
The vertices of this quadrilateral are O = (0,0), = (0,12), = (6,10), = (11,0).
The coordinates are easily found as the intersections of the corresponding lines.
Finding these coordinates is traditionally considered as elementary operations for LP problems,
so I will not bore the reader with these details.
From here she goes on to solve the problem using the standard corner points method, in which the objective function is evaluated at each corner point of the feasibility region.
But it is not necessary to evaluate the objective function at every corner of the feasibility region. The corner point where the objective function is maximized can be determined by comparing the slopes of the constraint lines to the slope of the objective function. The objective function will be maximized where a line with the slope of the objective function just touches the feasibility region.
The slopes of the constraint lines are -2 and -1/3; the slope of the objective function is -8/13. Since -8/13 is between -2 and -1/3, the objective function will be maximized at the intersection of the two constraint lines -- at P2(6,10).
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