A piece of cardboard measure 12ft by 12ft. Corners are to be cut from it as shown by the broken lines, and the sides folded up to make a box with an open top. What size corners should be cut from the cardboard to make a boxx with the greatest possible volume.

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To find the size of the corners that should be cut from the cardboard to make a box with the greatest possible volume, we need to follow these steps:

1. Identify the dimensions: In this case, the cardboard measures 12ft by 12ft.

2. Determine the size of the corners to be cut: Let's assume that the size of each corner to be cut is denoted by 'x'. Since all four corners will be cut, the final dimensions of the cardboard will be (12-2x) by (12-2x).

3. Calculate the volume: The volume of the box can be calculated by multiplying the length (12-2x) by the width (12-2x) by the height (x). Therefore, the volume function can be expressed as V(x) = x(12-2x)(12-2x).

4. Simplify the volume function: Expand the equation V(x) = x(12-2x)(12-2x) to get V(x) = 4x^3 - 48x^2 + 144x.

5. Find the critical points: To find the critical points, take the derivative of the volume function with respect to 'x' and set it equal to zero. In this case, dV(x)/dx = 12x^2 - 96x + 144.

Solving 12x^2 - 96x + 144 = 0 gives us two critical points: x = 2 and x = 6.

6. Evaluate the function at the critical points and endpoints: Calculate the volume of the box at x = 0 (no corners cut), x = 2, x = 6, and x = 6 (maximum allowed size).

V(0) = 0, V(2) = 192, V(6) = 0.

7. Determine the maximum volume: Since V(2) = 192 is greater than the volumes at the endpoints, the maximum volume is achieved when x = 2.

Therefore, the size of the corners that should be cut from the cardboard to maximize the volume of the box is 2ft.