Question 1153924
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<table border = "1" cellpadding = "5">
<tr><td></td><td>Trick Ski</td><td>Slalom Ski</td><td>Max Labor Hrs Available Per Day</td></tr>
<tr><td>Fabricating</td><td>8</td><td>6</td><td>264</td></tr>
<tr><td>Finishing</td><td>1</td><td>1</td><td>40</td></tr>
</table>


x = number of trick skis sold
y = number of slalom skis sold
both x and y are nonnegative, so {{{x >= 0}}} and {{{y >= 0}}}


From the first row of the table we see that {{{8x+6y <= 264}}} because the 8x represents the labor hours for making x number of trick skis, while 6y is the labor hours for making y number of slalom skis. The total (8x+6y) cannot exceed 264 hours, which is why we end up with {{{8x+6y <= 264}}}


Through similar reasoning, the second row gives us this inequality: {{{x+y <= 40}}}



Graph the following system of inequalities
{{{system(x>=0, y>=0, 8x+6y<=264, x+y <=40)}}}
to get this
<img src = "https://i.imgur.com/IdBNIlh.png">
Let me know if you need me to go into further detail about this part.


The vertex points are:
A = (0, 0)
B = (0, 40)
C = (12, 28)
D = (33, 0)
Each vertex point is found by intersecting the boundary lines. For example, point C is the intersection of the boundary lines 8x+6y = 264 and x+y = 40. 


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"the profit on a trick ski is ​$40 and the profit on a slalom ski is ​$50​"

40x = profit from only the trick skis (sold at $40 each)
50y = profit from only the slalom skis (sold at $50 each)
P = total profit (both types of skis)
P = 40x + 50y



Plug each vertex point into the profit function


Plug in (x,y) = (0,0)
P =  40x + 50y
P =  40(0) + 50(0)
P = 0
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Plug in (x,y) = (0,40)
P =  40x + 50y
P =  40(0) + 50(40)
P = 2000
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Plug in (x,y) = (12,28)
P =  40x + 50y
P =  40(12) + 50(28)
P = 1880
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Plug in (x,y) = (33,0)
P =  40x + 50y
P =  40(33) + 50(0)
P = 1320
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Summary:
Min is P = 0 which occurs at (x,y) = (0,0)
Max is P = 2000 which occurs at (x,y) = (0,40)



To get the max daily profit of $2000 per day, you need to make x = 0 trick skis per day and y = 40 slalom skis per day.


Side note: If both x and y must be greater than zero, then the next highest profit possible is when (x,y) = (12,28). This profit is $1880 per day.
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