A customer has placed an order for some precision-engineered cubes, but part of the order was destroyed in a paint-spilling accident. The only information we have is that each cube has a surface area of 𝑥 centimeters squared and a volume of two 𝑥 centimeter cubed. What is the length of the side of each cube?
So what we know about this question is that we’re looking at cubes, and what we’re trying to find is the length of the side of each cube. So we call this 𝐿. Well, first of all, if we’re looking at the surface area of a cube, what this is equal to is six multiplied by 𝐿 squared. And that’s because if we have the area of one of the faces, then this is just gonna be 𝐿 multiplied by 𝐿, which is 𝐿 squared. And because there are six faces, it’s gonna be six 𝐿 squared. And the volume is going to be 𝐿 cubed. And that’s cause the volume of a cube is just the length multiplied by the length multiplied by the length, which is 𝐿 cubed.
So therefore, using information we’ve got in the question, we can say that 𝑥 is equal to six 𝐿 squared. And that’s because the surface area is 𝑥 centimeters squared. And then, what we can do is multiply this by two, so each side of the equation. And we’re gonna get two 𝑥 is equal to 12𝐿 squared. But why have we done this? Why do we want to know what two 𝑥 is equal to?
Well, we’ve done this because what we’re also told is that the volume is equal to two 𝑥. Well therefore, we can use the fact that the volume is equal to 𝐿 cubed to say that two 𝑥 is equal to 𝐿 cubed. So what we now have are two values for 𝑥, 12𝐿 squared or 𝐿 cubed. So what this means is that we can equate our values for two 𝑥. So we can say that 𝐿 cubed is equal to 12𝐿 squared. So then, if we divide through by 𝐿 squared, what we’re gonna get is 𝐿 is equal to 12. So therefore, what we can say is that the length of the side of each of the cubes is going to be 12 centimeters.