Question 1198953
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x = number of acres of potatoes
y = number of acres of corn
These are real numbers such that x ≥ 0 and y ≥ 0


Since the farmer has at most 70 total acres to work with, we can say:
x+y ≤ 70


20x = cost of just the potatoes
60y = cost of just the corn
20x+60y = total cost ≤ 3000
20x+60y ≤ 3000
20(x+3y) ≤ 3000
x+3y ≤ 3000/20
x+3y ≤ 150


System of inequalities
{{{system(x >= 0, y >= 0, x+y <= 70, x+3y <= 150)}}}


Use a graphing tool (such as Desmos or GeoGebra), or do so by hand, to create a graph like this
*[illustration Screenshot_177.png]
The blue shaded region represents the set of (x,y) points that satisfy all of the inequalities mentioned in the system above.
Points on the boundary are included in the shaded solution set.


The corner points are:
A = (0, 50)
B = (30, 40)
C = (70, 0)
D = (0, 0)
Each corner can be found using algebraic methods to solve systems of equations.
For example, use algebra to solve the system 
{{{system(x+y =70,x+3y=150)}}} 
to determine the location of corner point B(30,40).


Once we have these corner points established, we plug each of them into the profit function
P(x,y) = 150x+50y
where,
150x = profit from just the potatoes only ($150 per acre)
50y = profit from just the corn only ($50 per acre)


You should find the following<table border = "1" cellpadding = "5"><tr><td>Point</td><td>Coordinates</td><td>Profit</td></tr><tr><td>A</td><td>(0,50)</td><td>$2500</td></tr><tr><td>B</td><td>(30,40)</td><td>$6500</td></tr><tr><td>C</td><td>(70,0)</td><td>$10500</td></tr><tr><td>D</td><td>(0,0)</td><td>$0</td></tr></table>
Point C is the winner in terms of max profit.
This makes sense because the profit on potatoes is three times as much as corn. 


Therefore, the farmer should use all 70 acres to plant nothing but potatoes. 


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<font color=red>Answers:</font>
Acres of potatoes = <font color=red>70</font>
Acres of corn = <font color=red>0</font>
Max Profit = <font color=red>$10,500</font>
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