A solid metal cuboid whose dimensions are two 𝑥 centimeters, six 𝑥 centimeters, and 10𝑥 centimeters was melted and made into small cubes. If the edges of the small cubes are two 𝑥 centimeters, how many can be made from the melted cuboid?
So, what we’re gonna do is look at the cuboid and cubes separately. So first of all, what we want to do is find out the volume of each of them. Well, the volume of a cuboid is equal to the length multiplied by the width multiplied by the height. And the volume of a cube is equal to the length cubed, and that’s because all of the lengths are the same. But if we take a look at the cuboid, dimensions are two 𝑥, six 𝑥, and 10𝑥. So therefore, the volume is gonna be these dimensions all multiplied together, so two 𝑥 multiplied by six 𝑥 multiplied by 10𝑥. So what this is gonna give is 120𝑥 cubed. And that’s because two multiplied by six is 12 multiplied by 10 is 120. Then 𝑥 multiplied by 𝑥 multiplied by 𝑥 is 𝑥 cubed. And the units for this would be centimeters cubed.
And now if we move on to the cube, the volume of the cube is equal to two 𝑥 cubed. And then this again will give us eight 𝑥 cubed. And then once again, the units would be centimeters cubed. But we don’t need to really use them at the moment because we don’t need them for this step. Well, it’s worth noting here that the answer was eight 𝑥 cubed for the volume. But be careful. A common mistake is to put two 𝑥 cubed. And that’s because students forget to cube the two as well as the 𝑥.
Okay, so now we know the volume of the cuboid and the volume of the cube. What is the next step to solving this problem? Well, what we want to do now is work out the number of cubes that could be made from the melted cuboid. So to do that, what we’re gonna do is divide the volume of the cuboid by the volume of the cube. So we’ve got 120𝑥 cubed over eight 𝑥 cubed. Well, first of all, we can see that we got 𝑥 cubed in the numerator and denominator, so we can divide through by 𝑥 cubed. Well, what this leaves us with is 120 over eight. Well, 120 divided by eight is 15. So therefore, we can say that 15 cubes could be made from the melted cuboid.