Question 1198959
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Given Facts:<ol><li>No more than 50 beds of Type A can be made.</li><li>No more than 40 beds of type B can be made.</li><li>At least 60 beds in all must be made.</li><li>The maximum number of beds that can be produced is 80.</li><li>The profit on type A is Php300.</li><li>The profit on type B is Php150. </li></ol>x = number of beds of type A
y = number of beds of type B
where x,y are nonnegative integers


Fact 1 leads to the inequality {{{x <= 50}}}
Since x is nonnegative, we can further clarify that {{{0 <= x <= 50}}} (i.e. x is between 0 and 50 inclusive of each endpoint)
x is in the set {0,1,2,...,49,50}


Fact 2 leads to {{{y <= 40}}}, then we can clarify to get {{{0 <= y <= 40}}}
y is in the set {0,1,2,...,39,40}


Fact 3 means {{{x+y >= 60}}} because "at least 60" means "60 or more".


Then fact 4 says {{{x+y <= 80}}} to put a ceiling on the production amount. 
The most beds that can be made is 80.


Fact 5 tells us 300x is the profit for just the type A beds.
Fact 6 tells us 150y is the profit for just the type B beds.
Combine those facts to get 300x+150y as the total profit for both beds.


The goal is to max out P = 300x+150y


System of inequalities
{{{system(0 <= x <= 50, 0 <= y <= 40,x+y >= 60,x+y<=80)}}}
Graph
*[illustration Screenshot_179.png]
The blue trapezoidal region represents the set of (x,y) points that satisfy all of the inequalities mentioned in the system above.
Points on the boundary are part of the shaded solution set.


The corner points are
A = (20, 40)
B = (40, 40)
C = (50, 30)
D = (50, 10)
Each corner point can be determined using algebra.
For instance, intersect the line y = 40 and x+y = 60 to determine the location of point A(20,40)


After getting those corner points, we then will plug each into the profit function to see which yields the largest value of P.


If we tried the x and y coordinates of point A, then,
P = 300x+150y
P = 300*20+150*40
P = 12000
Repeat for points B through D


These are the results you should get<table border = "1" cellpadding = "5"><tr><td>Point</td><td>Coordinates</td><td>Profit (Php)</td></tr><tr><td>A</td><td>(20,40)</td><td>12000</td></tr><tr><td>B</td><td>(40,40)</td><td>18000</td></tr><tr><td>C</td><td>(50,30)</td><td>19500</td></tr><tr><td>D</td><td>(50,10)</td><td>16500</td></tr></table>Point C is the winner in terms of max profit of Php19500
The min profit happens at point A.


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Answer:


How many beds on both types must be produced to maximize the profit? <font color=red>50 of type A and 30 of type B</font>


What is the minimum profit?  <font color=red>Php 12000</font>


Edit: just in case you made a typo and are asking for the maximum profit, then that max profit would be <font color=red>Php 19500</font>
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