QuestionMinimize $c=x+2y$ with constraints: $x+4y \geq 23$ , $6x+y \geq 23$ , $x \geq 0$ , $y \geq 0$ .

Studdy Solution

STEP 1

Assumptions1. The objective function is $c=x+y$ which we need to minimize. The constraints are $x+4y \geq23$ , $6x+y \geq23$ , and $x \geq0, y \geq0$
3. The variables $x$ and $y$ are non-negative

STEP 2

We need to graph the constraints to find the feasible region. The feasible region is the set of points that satisfy all the constraints.

STEP 3

First, let's graph the constraint $x+y \geq23$ . To do this, we first treat the inequality as an equation and find the intercepts.For the x-intercept, set $y=0$ and solve for $x$ $x+(0) =23 \Rightarrow x =23$ For the y-intercept, set $x=0$ and solve for $y$ $0+y =23 \Rightarrow y = \frac{23}{} =5.75$

STEP 4

Next, let's graph the constraint $6x+y \geq23$ . Again, we first treat the inequality as an equation and find the intercepts.For the x-intercept, set $y=0$ and solve for $x$ $6x+0 =23 \Rightarrow x = \frac{23}{6} \approx3.83$ For the y-intercept, set $x=0$ and solve for $y$ $0+y =23 \Rightarrow y =23$

STEP 5

Now, let's graph the constraints $x \geq0$ and $y \geq0$ . These are simply the x and y axes.

STEP 6

The feasible region is the area where all the constraints overlap. This region is bounded by the lines we graphed and the axes.

STEP 7

The minimum of the objective function $c=x+2y$ will occur at one of the vertices of the feasible region. We need to find the coordinates of these vertices.

STEP 8

The vertices are the points of intersection of the lines $x+4y=23$ and $6x+y=23$ . We can find these points by solving these equations simultaneously.

STEP 9

Substitute $x=23-4y$ from the first equation into the second equation $6(23-4y) + y =23$

STEP 10

olve the equation for $y$ $138 -24y + y =23$ $-23y = -115$ $y = \frac{115}{23} =5$

STEP 11

Substitute $y=5$ into the first equation to find $x$ $x +4(5) =23$ $x =23 -20 =3$ So one vertex of the feasible region is $(3,5)$ .

STEP 12

The other vertices of the feasible region are the intercepts of the lines and the axes, which are $(23,0)$ , $(0,5.75)$ , and $(.83,0)$ .

STEP 13

Now we substitute these vertices into the objective function $c=x+2y$ to find which one gives the minimum value.

STEP 14

Substitute $(3,)$ into the objective function $c =3 +2() =13$

STEP 15

Substitute $(23,0)$ into the objective function $c =23 +2(0) =23$

STEP 16

Substitute $(0,5.75)$ into the objective function $c =0 +2(5.75) =11.5$

STEP 17

Substitute $(3.83,0)$ into the objective function $c =3.83 +2(0) =3.83$

STEP 18

The minimum value of the objective function is $3.83$ , which occurs at the vertex $(3.83,0)$ .
So the minimum value of $c$ is $3.83$ .

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