To find the maximum area of the rectangle, we need to find the points on the positive x-axis and the lines y= 4x and y= -5x + 6 where the rectangle's vertices can be formed.
Let's start by finding the points on the positive x-axis. Since the positive x-axis lies on the y=0 line, the two points on the positive x-axis for the rectangle would be (x, 0) and (x', 0), where x and x' are the x-coordinates of the vertices we want to find.
Next, we need to find the points on the line y=4x and y=-5x+6 where the rectangle's vertices exist. These points can be found by substituting y=4x and y=-5x+6 into the equation of the positive x-axis.
For the point (x, 0):
0 = 4x
x = 0
For the point (x', 0):
0 = -5x' + 6
x' = 6/5
So, the points for the rectangle's vertices are (0, 0), (0, 0), (0, 0), and (6/5, 0), or simply (0, 0) and (6/5, 0).
Now, we can calculate the maximum area of the rectangle by finding the distance between these two points on the positive x-axis and multiplying it by the width of the rectangle, which is the difference between the y-coordinates of the two lines.
The distance between (0, 0) and (6/5, 0) is (6/5 - 0) = 6/5.
The width of the rectangle can be found by subtracting the y-coordinates of the two lines:
y=4x: 4x = 0 => x = 0 => y = 4(0) = 0
y= -5x + 6: -5x + 6 = 0 => -5x = -6 => x = 6/5 => y = -5(6/5) + 6 = 0
Therefore, the width of the rectangle is 0 - 0 = 0.
Lastly, we can calculate the maximum area of the rectangle by multiplying the distance and the width:
Area = Distance Γ Width = (6/5) Γ 0 = 0.
Therefore, the maximum area of the rectangle is 0.