Question 1066773
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Carlo and Anita make mailboxes and toys in their craft shop near Lincoln. 
Each mailbox requires 3 hours of work from Carlo and 4 hours from Anita. 
Each toy requires 2 hours of work from Carlo and 4 hours from Anita. 
Carlo cannot work more than 18 hours per week and 
Anita cannot work more than 32 hours per week. 
If each mailbox sells for $11 and each toy sells for $12, 
then how many of each should they make to maximize their revenue? What is their maximum revenue?
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I re-formatted the input to provide the maximal readability (!!)


<pre>
The question is: how many mailboxes (X) and how many toys (Y) should be produced to maximize the revenue $11*X + $12*Y
under these restrictions:

3X + 2Y <= 18     (1)     (Carlo restricted by 18 hours per week) and
4X + 4Y <= 32     (2)     (Anita restricted by 32 hours per week).

In other words, you must maximize the objective function (revenue) 

R(X,Y) = 11X + 12Y

over the domain on the plot below, which is  a quadrilateral in the first quadrant (X >= 0,  Y >= 0) restricted 
by the red and the green lines.


{{{graph( 330, 330, -2.5, 10.5, -2.5, 10.5,
          (18-3x)/2,  (32-4x)/4
)}}}


Plots y = {{{(18-3x)/2}}}  (red) and y = {{{(32-4x)/4}}} (green)



The method of linear programming says:

&nbsp;&nbsp;&nbsp;&nbsp;1) Take the vertices of this quadrilateral

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(x1,Y1) = (0,8)   (green line Y-intercept)
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(x2,Y2) = (6,0)   (red line X-intercept)
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(x3,Y3) = (2,6)   (intersection point of the straight lines Y = {{{(18-3x)/2}}} and Y = {{{(32-4x)/4}}} )

&nbsp;&nbsp;&nbsp;&nbsp;2) Calculate the objective function at these points

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;R(X1,Y1) = 11*0 + 12*8 = 96;

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;R(X2,Y2) = 11*6 + 12*0 = 66;

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;R(X3,Y3) = 11*2 + 12*6 = 94.


&nbsp;&nbsp;&nbsp;&nbsp;3) Then select one of these point where the objective function is maximal - In our case this point is (X1,Y1) = (0,8)


&nbsp;&nbsp;&nbsp;&nbsp;4) This point gives your optimal solution X = 0 mailboxes and Y = 8 toys.


If they follow this optimal solution, their weekly revenue will be MAXIMAL, $96.
</pre>

Solved.