Question

Question #37a3d

Answer 1

`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:

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`