SOLUTION: HELP PLEASE!!! Need to make both airplanes and helicopters Can't make more than 50 airplanes and no more than 30 helicopters Cost to make airplane - 60,000 Cost to make helicop

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Question 88286: HELP PLEASE!!!
Need to make both airplanes and helicopters
Can't make more than 50 airplanes and no more than 30 helicopters
Cost to make airplane - 60,000
Cost to make helicopter - 90,000
Budget to spend - 3.6 million
Need to find Max. profit if get 400,000 for each sirplane sold amd 850,000 for each helicopter sold???
Also - how many helicopters cam be made
HELP
NEED:
Constraints, object function, and solution
thanks so much
lucyloo
Found 2 solutions by stanbon, Edwin McCravy:
Answer by stanbon(75887) About Me  (Show Source):
You can put this solution on YOUR website!
Need to make both airplanes and helicopters
Can't make more than 50 airplanes and no more than 30 helicopters
a<=50 ; h<=30
----------------------
Cost to make airplane - 60,000
Cost to make helicopter - 90,000
Budget to spend - 3.6 million
60000a + 90000h<= 3.6 million
--------------------------------
Need to find Max. profit if get 400,000 for each sirplane sold amd 850,000 for each helicopter sold???
Profit on each airplane = 400,000-60,000 = 340,000
Profit on each helicopter = 850,000-90,000= 760,000
Object function: Profit = 340,000a + 760000h
-------------------------
Also - how many helicopters cam be made
HELP
NEED:
Constraints, object function, and solution
-------------------------------
You need to draw a coordinate system with axis "a" and "h"
Graph the constraints listed above.
Find the coordinates of the intersection points of the constraints.
Test each of those intersection points coordinate pairs in the
object function to see which gives you the maximum profit.
=========================
Cheers,
Stan H.

Answer by Edwin McCravy(20054) About Me  (Show Source):
You can put this solution on YOUR website!
Need to make both airplanes and helicopters
Can't make more than 50 airplanes and no more than 30 helicopters
Cost to make airplane - 60,000
Cost to make helicopter - 90,000
Budget to spend - 3.6 million
Need to find Max. profit if get 400,000 for each sirplane sold amd 850,000 for each helicopter sold???
Also - how many helicopters cam be made
HELP
NEED:
Constraints, object function, and solution
thanks so much
lucyloo

Let x = the number of airplanes to make.
Let y = the number of helicopters to make.

>>...Can't make more than 50 airplanes...<<

So that says

x < 50

>>...no more than 30 helicopters...<<

y < 30

>>...Cost to make airplane - 60,000...<<

That says 

the cost to make the x airplanes is 60000x

>>...Cost to make helicopter - 90,000...<<

That says:

The cost to make y helicopters is 90000y

>>...Budget to spend - 3.6 million...<<

That says 

60000x + 90000 < 3600000

So the constraints are:

              x < 50
              y < 30
60000x + 90000y < 3600000
              x > 0
              y > 0

Those last two are to prevent the number of 
planes or helicopters from being negative. 

We can simplify 

 60000x + 90000 < 3600000

by dividing through by 30000, getting

        2x + 3y < 120

So the constraints are

      x < 50
      y < 30
2x + 3y < 120
      x > 0
      y > 0

>>...get 400,000 for each sirplane sold...<<
>>...Cost to make airplane - 60,000...<<

So the profit made on an airplane is 
400000 - 60000 = 340000

So the profit made on x airplanes is 340000x

>>...850,000 for each helicopter sold...<<
>>...Cost to make helicopter - 90,000...<<

So the profit made on a helicopter is 
850000 - 90000 = 760000

So the profit made on y helicopters is 760000x

So the total profit P for making x airplanes and
y helicopters is

P = 340000x + 760000y

So to state the problem correctly:

Maximize the objective function

P = 340000x + 760000y

subject to the constraints:

      x < 50
      y < 30
2x + 3y < 120
      x > 0
      y > 0

We draw the graphs of these equations, formed
by replacing the symbols of inequality by
equal signs:

      x = 50  (a vertical line 50 units right of the y-axis)
              (the feasible region is on or left of this line) 
      y = 30  (a horizontal line 30 units above the x-axis
              (the feasible region is on or below this line) 
2x + 3y = 120 (a slanted line with intercepts (60,0), (0,40)
              (the feasible region is on or between this line
               and the x- and y- axes.  
      x = 0   (the y-axis)
              (the feasible region is on or right of the y-axis 
      y = 0   (the x-axis)
              (the feasible region is on or above the x-axis) 
 


The feasible region is this polygon:



Find the corner points by substitution using the 
equations:



Now substitute each of those corner points in the
objective function and you'll find that the optimum
point is (15,30) for a profit of $27,900,000 when
they make 15 airplanes and 30 helicopters.

Edwin