Question 84764
<pre><font size = 5><b>
The maximum value of z = 5x + 4y subject to 

3x + y <u><</u> 24
6x + 4y <u><</u> 66
x <u>></u> 0, y <u>></u> 0 is 

A. 96
B. 66
C. 56
D. 40

Graph the boundary lines:

1.     3x + y = 24  (3x + y <u><</u> 24 will 
be the region on or below this line)

{{{drawing(218.75,500,-2,12,-2,30,

 graph(218.75,500,-2,12,-2,30), line (-2,30,9,-3) )}}}

2.     6x + 4y = 66  (6x + 4y <u><</u> 66 will 
be the region on or below this line)

{{{drawing(218.75,500,-2,12,-2,30,

 graph(218.75,500,-2,12,-2,30), line(-2,30,9,-3),
 line(-2,19.5,12.33333,-2) 

 )}}}

3.     x = 0  (x <u>></u> 0 will be the 
region on or to the right of this line, 
which is just the y-axis.

{{{drawing(218.75,500,-2,12,-2,30,

 graph(218.75,500,-2,12,-2,30), line(-2,30,9,-3),
 line(-2,19.5,12.33333,-2), line(0,-2,0,30) 

 )}}}  

4.     y = 0  (y <u>></u> 0 will be the 
region on or above this line, which is 
just the x-axis.

{{{drawing(218.75,500,-2,12,-2,30,

 graph(218.75,500,-2,12,-2,30), line(-2,30,9,-3),
 line(-2,19.5,12.33333,-2), line(0,-2,0,30), line(-2,0,12,0) 

 )}}} 

You can shade the common region.  I can't shade 
on here so I will just erase all the parts of 
the lines that I don't need:

{{{drawing(218.75,312.5,-2,12,-2,18,

 graph(218.75,312.5,-2,12,-2,18), line(0,16.5,5,9),
 line(5,9,8,0), line(0,0,0,16.5), line(0,0,8,0) 

 )}}}

Now we will find all four corner points.

The top point is found by solving the system

6x + 4y = 66
x = 0

That has the solution (0,16.5)

The bottom left point is obviously the 
origin but is found by solving the system

x = 0
y = 0

That has solution (0,0)

The bottom right point is found by solving
the system

3x + y = 24
y = 0 

That has solution (8,0)

The point in the middle is found by solving 
the system

3x + y = 24
6x + 4y = 66

That has solution (5,9)

{{{drawing(218.75,312.5,-2,12,-2,18,

 graph(218.75,312.5,-2,12,-2,18), line(0,16.5,5,9),
 line(5,9,8,0), line(0,0,0,16.5), line(0,0,8,0),
  locate(5.5,10,"(5,9)"), locate(1,17,"(0,16.5)"),
  locate(8,1.5,"(8,0)"), locate(.5,1.5,"(0,0)") 

  )}}}

Now both the maximum and the minimum values of
the objective function 

z = 5x + 4y

will occur at corner points. So we make this table:

corner point |  x  |  y  |    z = 5x + 4y   |
---------------------------------------------
   (0,16.5)  |  0  |16.5 |  5(0)+4(16.5)=66   
    (0,0)    |  0  |  0  |  5(0)+4(0) =   0
    (8,0)    |  8  |  0  |  5(8)+4(0) =  40
    (5,9)    |  5  |  9  |  5(5)+4(9) =  61

So we find that the maximum value of the
objective function z is 66 when x=0 and y=16.5
(and the minimum value is 0 when x=0 and y=0).

But you wanted the maximum value so it's

z = 66 when x=0 and y = 16.5, which is
choice B.

Edwin</pre>