Question 1199207
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Given Facts:<ol><li>A company manufactures two products, A and B, and each of these products must be processed on two different machines. </li><li>Product A requires 2 min of work time per unit on machine 1 and 4 min of work time per unit on machine 2. </li><li>Product B requires 3 min of work time per unit on machine 1 and 1 min of work time per unit on machine 2. </li><li>Each day 200 min are available on machine 1 and 300 min are available on machine2. </li><li>To satisfy certain customers, the company must produce at least 8 units per day of product A and at least 10 units per day of product B. </li><li>The profit of each unit of product A is Php50 and the profit of each unit of product B is Php60.</li></ol>Let
x = number of units of product A
y = number of units of product B


Fact 6 tells us that the objective function is
P = 50x+60y
where P is the profit in Php.


Using fact 2 through fact 4, we can form this table to determine time constraints for each machine.<table border = "1" cellpadding = "5"><tr><td></td><td>Product A</td><td>Product B</td><td>Total</td><td>Constraint</td></tr><tr><td>Machine 1</td><td>2x</td><td>3y</td><td>2x+3y</td><td>{{{2x+3y <= 200}}}</td></tr><tr><td>Machine 2</td><td>4x</td><td>1y</td><td>4x+1y</td><td>{{{4x+1y <= 300}}}</td></tr></table>Each time value is in minutes.


The key thing to pull out of that table are these two constraints:
{{{2x+3y <= 200}}}
{{{4x+1y <= 300}}}


Another two constraints are:
{{{x >= 8}}}
{{{y >= 10}}}
Refer to fact 5.
Something like "at least 8" means "8 or more".


System of inequalities
{{{system(
2x+3y <= 200,
4x+1y <= 300,
x >= 8,
y >= 10
)}}}


Use graphing software or do so by hand to create the feasibility region.
I used GeoGebra to make the diagram shown below. The feasibility region is in blue.
*[illustration Screenshot_196.png]
The points on the boundary are part of the feasibility region.


It turns out that the objective function reaches its highest and lowest point at the corners of the feasibility region.

To find the location of each corner, use algebra.
For instance, the location of point B is found by solving this system
{{{system(2x+3y = 200,4x+y = 300)}}}
since B is where these two lines cross.


Here are the corner points:
A = (8, 61.3333)
B = (70, 20)
C = (72.5, 10)
D = (8, 10)
The y coordinate of point A is approximate.


Let's check each corner.


Checking point A
P = 50x+60y
P = 50*8+60*61.3333
P = 4079.998
P = 4080


Checking point B
P = 50x+60y
P = 50*70+60*20
P = 4700


Checking point C
P = 50x+60y
P = 50*72.5+60*10
P = 4225


Checking point D
P = 50x+60y
P = 50*8+60*10
P = 1000


The largest result for P was when we got P = 4700 (for point B)


Therefore, the largest profit possible is Php4700.
This profit is reached when 70 units of product A are made, and 20 units of product B are made.
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