I've got a sizable chunk of delicious sharp cheddar. My block of cheese is a perfect rectangular solid, dimensions are 10in x 13in x 14in.

PLease help

I have 10 friends at my house, so I cut 10 slices. Each slice is of width 1in, and it is cut parallel to one face of the cheese. The individual slices are not necessarily parallel to each other.

What is the maximum possible volume (in cubic in) of the remaining block after the 10 slices have been cut off?

1149

729

To find the maximum possible volume of the remaining block after cutting off the slices, we first need to determine how the slices are being cut. You mentioned that the slices are not necessarily parallel to each other, but they are parallel to one face of the cheese block.

Since each slice has a width of 1 inch, it means that the slices will be 1 inch thick. We can imagine that the slices are being cut either vertically or horizontally, depending on the orientation of the cheese block.

If we assume that the slices are being cut vertically, parallel to the 10-inch side of the cheese block, it means that each slice would have a length of 13 inches and a height of 14 inches. Therefore, each slice would have a volume of 1 in x 13 in x 14 in = 182 cubic inches.

In this case, the remaining block after cutting off all 10 slices would have dimensions of 10 inches x 13 inches x (14 inches - 10 slices x 1 inch) = 10 inches x 13 inches x 4 inches. The maximum possible volume of the remaining block would be 10 in x 13 in x 4 in = 520 cubic inches.

On the other hand, if we assume that the slices are being cut horizontally, parallel to the 14-inch side of the cheese block, each slice would have a length of 10 inches and a height of 13 inches. Therefore, each slice would have a volume of 1 in x 10 in x 13 in = 130 cubic inches.

In this case, the remaining block after cutting off all 10 slices would have dimensions of (10 inches - 10 slices x 1 inch) x 13 inches x 14 inches = 0 inches x 13 inches x 14 inches. Since the length of the remaining block would be zero, the maximum possible volume of the remaining block would also be zero.

Therefore, the maximum possible volume of the remaining block after cutting off all 10 slices would be 520 cubic inches when assuming the slices are being cut vertically, parallel to the 10-inch side of the cheese block.