<span>Constraints (in slope-intercept form) x≥0, y≥0, y≤1/3x+3, y</span>≤ 5 - x
The vertices are the points of intersection between the constraints, or the outer bounds of the area that agrees with the constraints. We know that x≥0 and y≥0, so there is one vertex at (0,0) We find the other vertex on the y-axis, plug in 0 for x in the function: y <span>≤ 1/3x+3 y </span><span>≤1/3(0)+3 y = 3. There is another vertex at (0,3) Find where the 2 inequalities intersect by setting them equal to each other (1/3x+3) = 5-x Simplify Simplify Simplify x = 3/2 Plugging in 3/2 into y = 5-x: 10/2 - 3/2 = 7/2 y=7/2 There is another vertex at (3/2, 7/2)
There is a final vertex where the line y=5-x crosses the x axis: 0 = 5 -x , x = 5 The final vertex is at point (5, 0) Therefore, the vertices are: (0,0), (0,3), (3/2, 7/2), (5, 0) We want to maximize C = 6x - 4y. Of all the vertices, we want the one with the largest x and smallest y. We might have to plug in a few to see which gives the greatest C value, but in this case, it's not necessary. The point (5,0) has the largest x value of all vertices and lowest y value. Maximum of the function: C = 6(5) - 4(0) C = 30</span>
