Question 1207808
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x = number of bags of popcorn
y = number of cans of soda


Since each bag of popcorn costs $2, she spends 2x dollars on popcorn alone.
Each can of soda costs $1, meaning she spends 1y = y dollars on soda alone.
2x+y = total amount spent for both items combined
This will set up the objective function C = 2x+y.
The goal is to make the output of this function the smallest possible, i.e. we want the smallest possible cost.


"<font color=blue>She only has room to store 24 cans of soda and 50 bags of popcorn at her house</font>" gives us these constraints {{{0 <= x <= 50}}} and {{{0 <= y <= 24}}}


"<font color=blue>She needs to have at least 60 items total to satisfy her guests</font>" gives this constraint {{{x+y >= 60}}}


Here are all of the constraints listed as a system of inequalities
{{{system(0 <= x <= 50,0 <= y <= 24,x+y >= 60)}}}


The first two inequalities define a rectangle that is 50 units across and 24 units tall. 
The lower left corner of this rectangle is at the origin (0,0). 
The upper right corner is at (50,24).


We'll overlap this rectangle with {{{x+y >= 60}}}
This is the region where we shade above the line x+y = 60. 
Two points on this line are (0,60) and (60,0)



Here's what the final region looks like
Any point in the purple region, or on the boundary, will satisfy all three constraint inequalities.
*[illustration UploadedScreenshot_54.png]
Graph created with Desmos.


Desmos link here
<a href="https://www.desmos.com/calculator/9svzgkteyb">https://www.desmos.com/calculator/9svzgkteyb</a>


The triangular region has the vertices:
(50,10)
(50,24)
(36,24)


These vertices are determined by intersecting the various boundary lines. 
For example, solve the system {x+y=60,y=24} to determine the vertex location (36,24).


The idea is to plug those coordinates into the objective function to figure out which x,y pair gives the smallest value of C.
Let's try the vertex (50,10)
C = 2x+y
C = 2*50+10
C = 110


Now try the vertex (50,24)
C = 2x+y
C = 2*50+24
C = 124
So far the first vertex leads to the lowest cost.


Now the last vertex
C = 2x+y
C = 2*36+24
C = 96
This is the smallest cost of the trio.



Therefore, the lowest cost ($96) occurs when Sophia buys x = 36 bags of popcorn and y = 24 cans of soda.
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