Final answer:
To find the maximum possible area of a rectangle with vertices on the graph of y = 12 - x^3/3 and the x-axis, set y = 0 and solve for x to find the x-coordinates of the upper vertices. Calculate the difference between the x-coordinates of the upper vertices and the y-coordinates of the lower vertices to find the dimensions of the rectangle. Multiply the length and width to find the maximum possible area and we get approximately 10.897 square units.
Step-by-step explanation:
To find the maximum possible area of the rectangle, we need to find the upper two vertices of the rectangle on the graph of y = 12 - x^3/3. Since the lower two vertices are on the x-axis, their y-coordinates are 0. We can set y = 12 - x^3/3 = 0 and solve for x to find the x-coordinates of the upper two vertices. Once we have the coordinates of all four vertices, we can determine the dimensions of the rectangle and calculate its area.
Let's solve for x:
12 - x^3/3 = 0
x^3 = 36
x = ∛36
x ≈ 3.301
Therefore, the x-coordinates of the upper two vertices are approximately 3.301. The y-coordinates are 0 since the lower vertices are on the x-axis.
The dimensions of the rectangle are the difference between the x-coordinates of the upper vertices and the y-coordinates of the lower vertices. The length is 3.301 - 0 = 3.301 and the width is also 3.301 - 0 = 3.301. The area is the product of the length and width: 3.301 * 3.301 = 10.897. Therefore, the maximum possible area of the rectangle is approximately 10.897 square units.