Question #37a3d

1 Answer
Oct 10, 2017

#x=8 and y=8# maximizes #z# so #z=40#

Explanation:

I’m going to use desmos.com to show what is going on, but you could do this by hand on graph paper as well:

Change the first two constraints so that they are inequalities of lines in slope intercept form:

#x+y>10#
#y> -x+10# (above but not equal to the line #y=-x+10#)

#x-y>0#
#y< x# (below but not equal to the line #y=x#)

We know the other constraints keeps us between 0 and 8 on the x-axis, while also not letting us go below 0 on the y-axis.

So this is what the graph looks like on Desmos:enter image source here

We can see the region we care about is the triangle that is contained in the first quadrant. This region has 3 vertices that could maximize our function #z#.

Let’s try each point with the function #z#:

#(8,8):2(8)+3(8)=40#

#(5,5):2(5)+3(5)=25#

#(8,2):2(8)+3(2)=22#

Thus the point #(8,8)# maximizes #z# so #x=8 and y=8#