Final answer:
The coordinates of the rectangle's vertices are: (0, 0), (6, 0), (0, 3), and (6, 3).
Step-by-step explanation:
The rectangle is bounded by the x-axis, y-axis, and the graph of y = (6−x)/2. To determine the coordinates of the rectangle's vertices, we need to find the points where the graph intersects the x and y-axes.
To find the x-intercept, we set y = 0 and solve for x:
0 = (6−x)/2
0 = 6−x
x = 6
So one of the vertices of the rectangle is (6, 0), where the graph intersects the x-axis.
To find the y-intercept, we set x = 0 and solve for y:
y = (6−0)/2
y = 3
So another vertex of the rectangle is (0, 3), where the graph intersects the y-axis.
The other two vertices of the rectangle are the origin (0, 0) and the point where the graph reaches its maximum or minimum value, which occurs at the vertex of the parabola. To find this point, we can take the derivative of y = (6−x)/2 and set it equal to zero:
y' = -1/2
-1/2 = 0
No solution
Since there is no solution, the graph does not have a minimum or maximum point, which means the other two vertices of the rectangle coincide with the origin and the point (6, 0).