QuestionSketch the graph for the constraints: $x \geq 50$ , $y \geq 60$ , $x+y \leq 200$ , $4x+5y \leq 900$ . Find the feasible region and minimize $P=0.3x+0.5y$ .

Studdy Solution

STEP 1

Assumptions1. The constraints are given as - $x \geq50$ - $y \geq60$ - $x + y \leq200$ - $4x +5y \leq900$ . The objective function is $=0.3x +0.5y$
3. We are looking for values of $x$ and $y$ that will minimize the profit4. The feasible region is the region that satisfies all the constraints

STEP 2

First, we need to sketch the constraints on a graph. We will start with the constraint $x \geq50$ . This is a vertical line at $x =50$ .

STEP 3

Next, we sketch the constraint $y \geq60$ . This is a horizontal line at $y =60$ .

STEP 4

The constraint $x + y \leq200$ is a line that passes through the points $(200,0)$ and $(0,200)$ . We sketch this line on the graph.

STEP 5

The constraint $4x +5y \leq900$ is a line that passes through the points $(225,0)$ and $(0,180)$ . We sketch this line on the graph.

STEP 6

Now, we need to find the feasible region. This is the region that satisfies all the constraints. It is the intersection of the regions defined by each constraint. In this case, it is the region above the line $y =60$ , to the right of the line $x =50$ , below the line $x + y =200$ , and below the line $4x +5y =900$ .

STEP 7

To find the values of $x$ and $y$ that will minimize the profit, we need to evaluate the objective function at each corner point of the feasible region. The corner points are the points where the boundary lines of the feasible region intersect.

STEP 8

The corner points of the feasible region are $(50,60)$ , $(50,150)$ , and $(125,75)$ .

STEP 9

We evaluate the objective function $=.3x +.5y$ at each corner point.

STEP 10

At $(50,60)$ , $=0.3(50) +0.5(60) =15 +30 =45$ .

STEP 11

At $(50,150)$ , $=0.3(50) +0.5(150) =15 +75 =90$ .

STEP 12

At $(125,75)$ , $=0.(125) +0.5(75) =37.5 +37.5 =75$ .

STEP 13

The minimum value of the objective function is $45$ at the point $(50,60)$ . So, the values of $x$ and $y$ that will minimize the profit are $x =50$ and $y =60$ .

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