Question 316557
Let x be the number of Product A.
Let y be the number of Product B.
The profit is then
{{{P(x,y)=50x+45y}}}
with the constraints that,
{{{x>=0}}}
{{{y>=0}}}
Assembly Time Total
{{{3x+4y<=2700}}}
{{{4y<=-3x+2700}}}
{{{y<=-(3/4)x+675}}}
Finishing Time Total
{{{2x+6y<=2400}}}
{{{6y<=-2x+2400}}}
{{{y<=-(1/3)x+400}}}
Plot all of the constraints to find the feasible region.
{{{graph(300,300,-200,1800,-200,800, -(3/4)x+675,-x/3+400)}}}
Find the intersection points.
(0,0)
(0,400)
(900,0)
and the intersection of the two lines,
1.{{{3x+4y=2700}}}
2.{{{2x+6y=2400}}}
Multiply eq. 1 by (-2) and eq. 2 by (3) and add them,
{{{-6x-8y+6x+18y=-5400+7200}}}
{{{10y=1800}}}
{{{highlight(y=180)}}}
Then from eq. 1,
{{{3x+720=2700}}}
{{{3x=1980}}}
{{{highlight(x=660)}}}
.
.
.
(660,180)
{{{drawing(300,300,-200,1800,-200,800, 
grid(1),
circle(660,180,43.3),
circle(0,0,43.3),
circle(0,400,43.3),
circle(900,0,43.3),

graph(300,300,-200,1800,-200,800,  -(3/4)x+675,-x/3+400))}}}
.
.
.
Check the profit function at each of the vertices.
The max and min will occur at one of these points.
(0,0):{{{P(x,y)=50x+45y=0+0=0}}}
(0,400):{{{P(x,y)=50x+45y=50(0)+45(400)=18000}}}
(900,0):{{{P(x,y)=50x+45y=50(900)+45(0)=45000}}}
(660,180):{{{P(x,y)=50x+45y=50(660)+45(180)=41100}}}
.
.
.
Maximum profit is $45,000 and occurs when you make 900 of Product A and 0 of Product B.