11. The area of a rectangle can be modeled by A=b(4-b), where A is the area of the rectangle and b is the length of the base (in inches). What is the greatest possible area of the rectangle?

Step 1 ：The problem is asking for the maximum area of a rectangle given the equation \(A = b(4-b)\). This is a quadratic equation, and the maximum or minimum of a quadratic equation can be found by finding the vertex of the parabola it represents.

Step 2 ：The x-coordinate of the vertex of a parabola given by the equation \(y = ax^2 + bx + c\) is \(-b/2a\). In this case, \(a = -1\) and \(b = 4\), so the x-coordinate of the vertex is \(-4/2(-1) = 2\). This means that the maximum area occurs when the base of the rectangle is 2 inches.

Step 3 ：To find the maximum area, we substitute \(b = 2\) into the equation for \(A\).

Step 4 ：Substituting \(b = 2\) into the equation gives \(A = 4\).

Step 5 ：Final Answer: The greatest possible area of the rectangle is \(\boxed{4}\) square inches.