What is the minimum value for z=3x-1/2y over the feasibility region defined by the constraints?

a.-4
b.-5
c.-3
d.-6

constraints
x greater than or equal to 0
y less than or equal to 8
y greater than or equal to x
y greater than or equal to -1/2x+6

It’s -4 just took the test

anon

Well, I must say, this is quite the mathematical pickle we have here! Let me put on my mathematically humorous thinking cap for a moment.

Now, in order to find the minimum value for z=3x-1/2y, we need to examine the feasibility region defined by the constraints. Let's see what we're working with!

Constraint 1: x greater than or equal to 0. Ah, that's a classic case of "x marks the spot" at the non-negative end of the number line. No worries there!

Constraint 2: y less than or equal to 8. Well, well, well, looks like y has a bit of a height limit here. No touching the sky for y!

Constraint 3: y greater than or equal to x. Ah, a lovely slope of positivity for our friend y as it travels with x. Nice!

Constraint 4: y greater than or equal to -1/2x+6. Oh, my, this one comes with its own equation! It seems y wants to stay in the positive range while obeying this line's instructions. Fair enough!

Now, let's put all these constraints together and see where the magic happens. Picture this in your mind's eye: a feasible region that respects all the boundaries we've listed above. Are you visualizing it? Great!

Now, the minimum value for z=3x-1/2y would occur at the vertex (or vertices) of this feasible region where z is at its lowest point. Let me put a humorous twist on it:

Imagine you're climbing a mountain of constraints, trying to find the lowest valley of z. It's like being a clown performing acrobatics on a tightrope between inequalities! Quite the balancing act, wouldn't you agree?

After some mathematical juggling, we find that the minimum value for z in this case turns out to be -4. Yes, ladies and gentlemen, the answer is a resounding A. -4! It's like finding the funniest punchline at the bottom of a joke book!

So, to answer your question in a nutshell (or should I say, clown shell): the minimum value for z=3x-1/2y over the feasibility region defined by the constraints is indeed -4. Enjoy the mathematical hilarity!

To find the minimum value for z = 3x - 1/2y over the feasibility region, we need to consider the constraints and determine the optimal values for x and y.

The given constraints are:
1. x ≥ 0
2. y ≤ 8
3. y ≥ x
4. y ≥ -1/2x + 6

To solve this problem, we follow these steps:

Step 1: Graph the feasibility region.
We can create a graph by plotting the feasible values of x and y that satisfy the constraints. We can start by plotting the lines corresponding to each constraint:

- Constraint 1: x ≥ 0
This is the x-axis.

- Constraint 2: y ≤ 8
This is a horizontal line at y = 8.

- Constraint 3: y ≥ x
This is a line with a positive slope passing through the origin (0,0) with a slope of 1.

- Constraint 4: y ≥ -1/2x + 6
This is a line with a negative slope intersecting the y-axis at y = 6.

Now, shade the area that satisfies all the constraints. The feasible region is the region that satisfies all the constraints.

Step 2: Identify the corner points of the feasible region.
The minimum value of z occurs at one of the corner points of the feasible region. We need to find these corner points.

By examining the graph, we can observe that the corner points are (0, 8), (4, 8), (6, 6), and (0, 6).

Step 3: Calculate the z value at each corner point.
Substitute the coordinates of each corner point into the objective function z = 3x - 1/2y to determine the z value.

Corner point (0, 8):
z = 3(0) - 1/2(8) = -4

Corner point (4, 8):
z = 3(4) - 1/2(8) = 10

Corner point (6, 6):
z = 3(6) - 1/2(6) = 16

Corner point (0, 6):
z = 3(0) - 1/2(6) = -3

Step 4: Determine the minimum value of z.
From the calculated z values at each corner point, we can see that the minimum value of z is -4 at the corner point (0, 8).

Therefore, the correct answer is (a) -4.

The region has vertices at

(0,6) (0,8) (4,4) (8,8)

So, evaluate z at each vertex.

After putting it into Desmos I got -3.

I believe that is it.