An engineering company makes some aluminum cubes for a customer. Each cube has a surface area of 𝑥 square centimetres and a volume of 𝑥 cubic centimetres. What is the length of the side of each cube?
We’re just going to begin by defining the side length of the cube. We’re going to let 𝑎 or 𝑎 centimetres be the side length of each cube. We’re told that the surface area of each cube is 𝑥 square centimetres. Now, of course, the surface area is the combined area of each of the faces. Each of the faces is a square, and so it has an area of 𝑎 times 𝑎, which is 𝑎 squared. A cube has six faces, so the total surface area of each cube will be six 𝑎 squared. But of course, we were told that the surface area was 𝑥, so we can say that 𝑥 must be equal to six 𝑎 squared.
Next, we’re told that the volume of each cube is 𝑥 cubic centimetres. Now, of course, the volume of a cuboid is the length multiplied by the base multiplied by the width. Now, since a cube is simply a cuboid whose lengths are all the same, we can say that the volume of our cube must be 𝑎 cubed. We were told that the volume is 𝑥, so we can say that 𝑥 is equal to 𝑎 cubed. Since we’ve said that 𝑥 is equal to six 𝑎 squared but also equal to 𝑎 cubed, this must mean that six 𝑎 squared equals 𝑎 cubed. And now, we have an equation in terms of 𝑎 that we can solve.
This is a cubic equation, so there could be up to three solutions. So, what we’re not going to do is divide through by 𝑎 squared. If we do that, we’re going to lose two solutions. Instead, we’re going to do what we do with a quadratic. We’re going to make equal to zero. So, we’re going to subtract six 𝑎 squared from both sides. And so, zero is equal to 𝑎 cubed minus six 𝑎 squared. We then factor the expression on the right-hand side. Now, each term has a common factor of 𝑎 squared. So, we get 𝑎 squared times 𝑎 minus six. And for 𝑎 squared times 𝑎 minus six to be equal to zero, either 𝑎 squared itself must be equal to zero or 𝑎 minus six must be equal to zero. We solve this first equation by square rooting both sides. And that gives us 𝑎 is equal to zero.
Now, of course, we can disregard this. 𝑎 is the length of a cube and the length of the cube cannot be zero. To solve the second equation, we’ll add six to both sides. And that gives us 𝑎 is equal to six. And so, we’re done. The side length of each cube is six centimetres. Now, in fact, it’s always sensible to check our answers. We check by substituting 𝑎 equals six into the expression six 𝑎 squared and 𝑎 cubed and checking we get the same answer. Six 𝑎 squared becomes six times six squared, which is 216. 𝑎 cubed is six cubed, which is also 216. We were hoping that these would be equal and they are, so we know what we’ve done is correct. The side length of each cube is six centimetres.