find the maximum value of the objective function and the values of x and y for which it occurs. F=2x+y
3x+5y<=45 x>=0 and y>=0
2x+4y<=32

Respuesta :

Answer:

The maximum value of the objective function is 30 at (15, 0).

Step-by-step explanation:

Given the objective function,  F = 2x + y

and the following constraints:

3x + 5y ≤ 45

2x + 4y ≤ 32

x ≥ 0  

y ≥ 0

You must first find the intercepts of the given constraints to plot the points on the graph, and find the feasible region.  

3x + 5y = 45

3x + 5y - 3x = - 3x + 45  

5y = -3x + 45

[tex]\frac{5y}{5} = \frac{-3x + 45}{5}[/tex]  

y = -3/5x + 9 (Slope-intercept form).  

Solve for y-intercept by setting x = 0:

y = -3/5x + 9

y = -3/5(0) + 9

y = 9

y-intercept: (0, 9).

Solve for x-intercept by setting y = 0:

y = -3/5x + 9

0 = -3/5x + 9

0 - 9 = -3/5x + 9 - 9

- 9 = -3/5x  

Multiply both sides by (-5/3) to isolate x:

[tex](-\frac{5}{3}) - 9 = (-\frac{5}{3}) - \frac{3}{5}x[/tex]

15 = x

x-intercept: (15, 0).

Next, transform 2x + 4y ≤ 32 into slope-intercept form, and solve for the y- and x-intercepts:

2x + 4y = 32

2x + 4y - 2x = - 2x + 32

4y = -2x + 32

Divide both sides by 4 to isolate y:  

[tex]\frac{4y}{4} = \frac{-2x + 32}{4}[/tex]

y = -1/2x + 8 (slope-intercept form)

Solve for y-intercept by setting x = 0:

y = -1/2x + 8

y = -1/2(0) + 8

y = 8

y-intercept: (0, 8).

Solve for x-intercept by setting y = 0:

y = -1/2x + 8

0 = -1/2x + 8

0 - 8 = -1/2x + 8 - 8

-8 = -1/2x

Divide both sides by -1/2 to solve for x:

-8 / -1/2 = (-1/2x)/ -1/2

[tex]\frac{-8}{- \frac{1}{2} } = \frac{-\frac{1}{2}x }{-\frac{1}{2}}[/tex]

16 = x

x-intercept: (16, 0).

Now that you have the following points: (0, 9), (15, 0), (0, 8), and (16, 0), you can plot these on the graph and setup a feasible region.  (Please see the attached screenshot for details.

The x-intercept, (16, 0) is not included because it is not within the feasible region.  

You must also evaluate the objective function at each corner point to identify which is the maximum value.

Corner Point              F = 2x + y

(0, 0)                      F = 2(0) + (0) = 0

(0, 8)                       F = 2(0) + (8) = 8

(0, 9)                      F = 2(0) + (9) = 9

(15, 0)                     F = 2(15) + (0) = 30

(10, 3)                      F = 2(10) + (3) = 23

Therefore, the maximum value of the objective function is 30 at (15, 0).  

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