The sum of three consecutive numbers is -12. Find the sum of the integers?
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This is a linear programming problem. When I say "programming", I don't mean computers. This can be done without the use of a computer though I don't recommend doing it by hand.
Let
x = number of trays of corn muffins
y = number of trays of bran muffins
Given info: "baking a tray of corn muffins takes 4 cups of milk and 3 cups of wheat flour"
If we focus on the milk only then baking x trays of corn muffins means we use up 4*x (or simply 4x) cups of milk. Focus now on the wheat flour and it takes 3*x or 3x cups of wheat flour
Also given: "Baking a tray of bran muffins takes 2 cups of milk and 3 cups of wheat flour"
So we use up 2y cups of milk and 3y cups of wheat flour
-----------------
So far we have used 4x cups of milk for the corn muffins plus 2y more cups of milk for bran muffins giving a total of 4x+2y cups of milk used. We only have 16 cups of milk which means we can say this
4x + 2y <= 16
where the "<=" symbol without quotes means "less than or equal to". That inequality above is one of the constraints placed.
-----------------
Another constraint is this inequality
3x+3y <= 15
because
* we use 3x cups of wheat flour for the corn muffins
* we use 3y cups of wheat flour for the bran muffins
* we have a max of 15 cups of wheat flour
-----------------
Additional contraints are:
x >= 0
y >= 0
indicating that we cannot have x or y be negative
-----------------
The four constraints are:
4x + 2y <= 16
3x+3y <= 15
x >= 0
y >= 0
If you graph all of these inequalities and do the proper shading, you'll get what you see in the attached image. Specifically I'm talking about the blue shaded region. This region is known as the feasible region.
Now turn your attention to the red corner points aka the vertices. These points will be plugged into the objective function
P(x,y) = 3x+2y
This represents the profit (in dollars) of making x trays of corn muffins and y trays of bran muffins.
3x = 3*x = profit from just the corn muffins only
2y = 2*y = profit from just the bran muffins only
3x+2y = total profit (selling both corn and bran muffins)
The four corner points are:
A = (3,2)
B = (0,5)
C = (0,0)
D = (4,0)
Plug each of these points into P(x,y) = 3x+2y
-----------------------------
Plug in A = (x,y) = (3,2)
P(x,y) = 3x+2y
P(3,2) = 3*3+2*2
P(3,2) = 13
We earn a profit of $13 if we sell 3 trays of corn muffins and 2 trays of bran muffins
-----------------------------
Plug in (x,y) = (0,5) which is point B
P(x,y) = 3x+2y
P(0,5) = 3*0+2*5
P(0,5) = 10
The profit dropped to $10, so we can ignore point B. So far point A leads to the most profit.
-----------------------------
Plug in point C, which is (x,y) = (0,0)
P(x,y) = 3x+2y
P(0,0) = 3*0+2*0
P(0,0) = 0
The profit drops even further. Ignore this point.
-----------------------------
Plug in point D: (x,y) = (4,0)
P(x,y) = 3x+2y
P(4,0) = 3*4+2*0
P(4,0) = 12
The profit has gone back up but not at the same height as $13 back for point A. So we can cross this off the list too
-----------------------------
The max profit happens at point A which again was
(x,y) = (3,2)
x = 3 ----> selling 3 trays of corn muffins
y = 2 ----> selling 2 trays of bran muffins
----------------------------------------------------------
----------------------------------------------------------
So the final answer is choice A "3 trays of corn muffins and 2 trays of bran muffins"
Let
x = number of trays of corn muffins
y = number of trays of bran muffins
Given info: "baking a tray of corn muffins takes 4 cups of milk and 3 cups of wheat flour"
If we focus on the milk only then baking x trays of corn muffins means we use up 4*x (or simply 4x) cups of milk. Focus now on the wheat flour and it takes 3*x or 3x cups of wheat flour
Also given: "Baking a tray of bran muffins takes 2 cups of milk and 3 cups of wheat flour"
So we use up 2y cups of milk and 3y cups of wheat flour
-----------------
So far we have used 4x cups of milk for the corn muffins plus 2y more cups of milk for bran muffins giving a total of 4x+2y cups of milk used. We only have 16 cups of milk which means we can say this
4x + 2y <= 16
where the "<=" symbol without quotes means "less than or equal to". That inequality above is one of the constraints placed.
-----------------
Another constraint is this inequality
3x+3y <= 15
because
* we use 3x cups of wheat flour for the corn muffins
* we use 3y cups of wheat flour for the bran muffins
* we have a max of 15 cups of wheat flour
-----------------
Additional contraints are:
x >= 0
y >= 0
indicating that we cannot have x or y be negative
-----------------
The four constraints are:
4x + 2y <= 16
3x+3y <= 15
x >= 0
y >= 0
If you graph all of these inequalities and do the proper shading, you'll get what you see in the attached image. Specifically I'm talking about the blue shaded region. This region is known as the feasible region.
Now turn your attention to the red corner points aka the vertices. These points will be plugged into the objective function
P(x,y) = 3x+2y
This represents the profit (in dollars) of making x trays of corn muffins and y trays of bran muffins.
3x = 3*x = profit from just the corn muffins only
2y = 2*y = profit from just the bran muffins only
3x+2y = total profit (selling both corn and bran muffins)
The four corner points are:
A = (3,2)
B = (0,5)
C = (0,0)
D = (4,0)
Plug each of these points into P(x,y) = 3x+2y
-----------------------------
Plug in A = (x,y) = (3,2)
P(x,y) = 3x+2y
P(3,2) = 3*3+2*2
P(3,2) = 13
We earn a profit of $13 if we sell 3 trays of corn muffins and 2 trays of bran muffins
-----------------------------
Plug in (x,y) = (0,5) which is point B
P(x,y) = 3x+2y
P(0,5) = 3*0+2*5
P(0,5) = 10
The profit dropped to $10, so we can ignore point B. So far point A leads to the most profit.
-----------------------------
Plug in point C, which is (x,y) = (0,0)
P(x,y) = 3x+2y
P(0,0) = 3*0+2*0
P(0,0) = 0
The profit drops even further. Ignore this point.
-----------------------------
Plug in point D: (x,y) = (4,0)
P(x,y) = 3x+2y
P(4,0) = 3*4+2*0
P(4,0) = 12
The profit has gone back up but not at the same height as $13 back for point A. So we can cross this off the list too
-----------------------------
The max profit happens at point A which again was
(x,y) = (3,2)
x = 3 ----> selling 3 trays of corn muffins
y = 2 ----> selling 2 trays of bran muffins
----------------------------------------------------------
----------------------------------------------------------
So the final answer is choice A "3 trays of corn muffins and 2 trays of bran muffins"