Problem number 26. When the height of a cuboid is reduced by three centimeters, its surface area is reduced by 60 centimeters squared. The resulting shape is a cube. What is the volume of the original cuboid in centimeters cubed? To solve this problem, let's think about the surface area that gets removed after we take off the top part. In particular, we know that the resulting shape will be a cube, so that means that let's say it has all sides of length x. Now, we can start to figure out some variables representing that surface area that is lost and gained. First of all, we know that the surface area that's lost includes the four lateral surfaces around the sides of the cuboid. Each of those lateral surfaces is a rectangle with height 3 and length x, so that is 4 times 3x, since each of these four faces has an area of 3x. We also lose the top part, which has an area of x times x, or x squared. But we actually gain an area of x squared too. Look at the region that's pointed to by the arrow. If we imagine cutting off the top part, we actually expose a new face of area x squared, because that face is originally covered up, but after we take off the top part, it gets exposed. So we actually gain an area of x squared too. This gained area of x squared actually cancels out the original area of x squared that's lost. So in fact, we only lose 4 times 3x units of area, that is, we lose 12x centimeters squared of area. Now we can solve for x. Since we know that we lose 60 centimeters squared of area, 12x equals 60, so x equals 5. Finally, we know the dimensions of the original cuboid. We know that the height is 5 plus 3, or 8, since we know that the height is 3 plus x, and now we know x is 5. Now we know all the dimensions. The cuboid is 8 by 5 by 5, so we can multiply those together and get 200. The original volume of the cuboid is D, 200.

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