A block of cheese 6cm by 8cm by 10cm is dipped into a vat of wax and covered on all sides. The cheese is then cut into 1cm cubes. How many of these cubes will not have any wax on them?
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Volume of the cuboid is 6 cm x 8 cm x 10 cm = 480 cm^3. That makes 480 little 1 cm^3 cubes. Now, let's take one side of the cuboid and observe it. For example, the one with the dimensions 6 cm x 8 cm. It is divided into 48 squares 1 cm^2 big. But, all the edges are covered with wax. That is, left and right side of the rectangle, and up and down. Because of this we reduce the length of the sides, each by two. This leaves us with the rectangle with the dimensions 4 cm x 6 cm. We have two sides on the cuboid like this. Let us take the second side of this cuboid, the one long 8 cm x 10 cm. Since we already reduced the 8 cm long sides of this to 6 cm, we are left with the 10 cm long ones. Now we reduce them too and we are left with two sides of the cuboid with the dimensions 6 cm x 8 cm. We do the same with the third one, and we get the dimensions of 4 cm x 8 cm. So, basically, we only reduced each side length by two, and now we have one with the dimensions 4 cm x 6 cm x 8 cm = 192 cm^3, which is at the same time the number of cubes. It is perhaps hard to picture this in your head, but I hope I didn't confuse you. The thing is, you only have to put off the first "layer" of the cuboid. You can draw this if it will be easier for you to understand. I hope I helped.
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Only the outside cubes will have wax on them. That means that you must remove 2 from each dimension, 1 for each side in the dimension, then multiply.
4cm x 6cm x 8cm
192cm^3
Each cube is 1cm^3, so there are 192 cubes that won't have wax on them.
4cm x 6cm x 8cm
192cm^3
Each cube is 1cm^3, so there are 192 cubes that won't have wax on them.
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