A sheet of cardboard 3ft by 4ft will be made into a box by cutting equal sized square
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To determine the size of the square to be cut from each corner of the cardboard sheet, we need to determine the dimensions of the resulting box.
Given that the cardboard sheet measures 3ft by 4ft, we know that the length of the box will be 3ft and the width of the box will be 4ft.
To find the dimensions of the resulting box, we subtract the lengths of the cut squares from the original dimensions of the cardboard sheet.
Let's assume that each side of the square to be cut has a length of "x". Therefore, when we cut out squares from all four corners, the resulting box will have a length of (3ft - 2x) and a width of (4ft - 2x).
To find the optimal size for the square to be cut, we need to determine the dimensions of the resulting box that maximizes its volume.
The volume of a rectangular box is determined by multiplying its length, width, and height. In this case, the height of the box will be "x" since the square cutouts will form the box's height.
Therefore, the volume of the box can be calculated as follows:
Volume = (3ft - 2x) * (4ft - 2x) * x
To maximize the volume, we can take the derivative of the volume equation with respect to "x" and set it equal to zero to find the critical points.
Once we find the critical points, we can determine which one corresponds to a maximum volume. However, this process involves calculus and may be lengthy for manual calculations.
Alternatively, we can use a graphical approach or technology such as a graphing calculator to plot the volume equation and find the maximum point by observing the shape of the graph.
Once the optimal value of "x" is found, we can substitute it back into the volume equation to determine the maximum volume of the resulting box.
Therefore, the process involves mathematical calculations and may be easier and faster to perform using appropriate tools or software.