Question 1185076
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The Halloween Express is making pre-packaged bags of candy for customers to give out to trick-or-treaters. 
The store has 20 tootsie rolls and 18 suckers. 
Package A requires 1 tootsie roll and 3 suckers. 
Package B requires 5 tootsie rolls and 1 sucker. 
If package A sells for $2 and package B sells for $3, how many of each package should they make to maximize {{{cross(profit)}}} <U>revenue</U> ?
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&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;It is quite obvious, &nbsp;that at the given circumstances, &nbsp;the problem's question 
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;should be about the &nbsp;REVENUE &nbsp;and not about the profit.


&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;With this editing, &nbsp;see my solution below.



<pre>
Let X be the number of packages A  and Y be the number of packages B.


Then we want maximize the revenue function

    R(X,Y) = 2X + 3Y


under the following restrictions

     X + 5Y <= 20      (toorsie rolls)

    3X +  Y <= 18      (suckers)

     X >= 0,  Y >= 0


The plot of the feasibility domain is shown in the Figure below.


    {{{graph( 400, 400, -2.5, 20.5, -2.5, 20.5,
          (20-x)/5,  18-3x
)}}}


    Plots y = {{{(20-x)/5}}}  (red) and y = 18-3x (green)



It is a quadrilateral in QI with the vertices  (X,Y) = (0,0), (0,4), (5,3), (6,0).


We apply the standard Linear Programming method in its geometric interpretation.


The solution is one of these 4 points, where the objective function (profit) has a maximum.


You calculate the values of the objective function  R(X,Y)  (revenue)  at listed points


    P(0,0) = 2*0 + 3*0 =    0

    P(0,4) = 2*0 + 3*4 =   12

    P(5,3) = 2*5 + 3*3 =   19

    P(6,0) = 2*6 + 3*0 =   12.


Comparing these values, you find the optimal point.

It is  (X,Y) = (5,3),  which means  5 boxes A and 3 boxes B, providing maximum REVENUE of 19 dollars.
</pre>

Solved.


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In this site, there is a lesson

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/word/misc/Solving-minimax-problems-by--the-Linear-Programming-method.lesson>Solving minimax problems by the Linear Programming method</A> 

which explains, for beginners, metodology of solving such problems in more details.