The minimum value of z = 4x = 10y subject to
3x + y <= 24
6x + 4y <= 66
x >= 0, y >= 0 is
a. 165
b. 110
c. 44
You mistyped when you typed
z = 4x = 10y
You must have either meant
z = 4x + 10y
or else you meant
z = 4x - 10 and pressed
So we draw the boundary lines from their equations
which we form by taking the inequalities and replacing
the inequality symbols by equal signs. So we draw
the graphs of
3x + y = 24
6x + 4y = 66
x = 0 (the y=axis)
y = 0 (the x-axis)
If we substitute a point say (1,1) in each original
inequality, we will see that the feasible region is
the region below the two slanted lines and the two
axes. That is this region:
If we solve the equations of the two slanted lines
we get that they intersect at the point (x,y) = (5,9)
That is one of the corner points of the region. The
other two corner points are
(x,y) = (0,0),
(x,y) = (0, 16.5), the y-intercept of 6x + 4y = 66
(x,y) = (8, 0), the x-intercept of 3x + y = 24.
Now if you meant z = 4x + 10y, then
corner point | x | y | 4x+10y |
--------------------------------
(0,0) | 0 | 0 | 0 |
(0, 16.5) | 0 | 16.5 | 165 |
(5, 9) | 5 | 9 | 110 |
(8, 0) | 8 | 0 | 32 |
----------------------------------
Then the minimum value of z is 0
(The maximum value of z is 165).
I'm starting to also suspect you may have
mistyped "minimum" for "maximum", for
0 is not listed though 165 is listed.
Another possibility is that you didn't mistype
"minimum" but you mistyped the inequality symbols
<=, when you meant >=, which
would have given the upper region
and then the corner points would have been
these, assuming you meant z = 4x + 10y, then
corner point | x | y | 4x+10y |
--------------------------------
(0, 24) | 0 | 24 | 240 |
(5, 9) | 5 | 9 | 110 |
(11, 0) |11 | 11 | 44 |
----------------------------------
Then the minimum value would have
been 44, which is listed
----------------------------------
Now if you meant z = 4x - 10y, then
corner point | x | y | 4x-10y |
--------------------------------
(0,0) | 0 | 0 | 0 |
(0, 16.5) | 0 | 16.5 | -165 |
(5, 9) | 5 | 9 | -70 |
(8, 0) | 8 | 0 | 32 |
----------------------------------
Then the minimum value of z is -165
The maximum value of z is 32. But
neither of those is listed, unless
you mistyped 165 for -165
What is sure is that you mistyped something,
but I can't tell what you mistyped. Maybe
you can figure out what you mistyped from
the above. If not, post again, this time
being extremely careful not mistype anything.
Edwin
