A metal rectangular prism has dimensions of nine centimeters, 12 centimeters, and two centimeters. If it is melted and converted to a cube, what will the length of the cube be?
So we have this metal rectangular prism with dimensions of nine centimeters, 12 centimeters, and two centimeters. And we’re taking this metal rectangular prism and we’re. melting it. And when we melt it, we’re converting it to a cube. And with a cube, all of the side lengths are equal in measure. And we want to find out what is this length.
So the side lengths are going to change when we go from a prism to a cube. However, what shouldn’t change is the volume, how much metal it’s actually made up of. So let’s find the volume of the rectangular prism: length times width times height.
So we needed to take two centimeters times 12 centimeters times nine centimeters and we get 216 centimeters cubed. And we know that the volume of a cube should be the exact same. Now the formula will stay the same: length times width times height. However, in a cube, all of the dimensions are the exact same. So we can call them all lengths.
So if we have length times length times length, we have the length cubed. And again, we know what this volume should be equal to: 216 centimeters cubed. So if we take 𝐿 cubed and set it equal to 216 centimeters cubed, we can solve for the length of the cube, which is we want.
So if we need to solve for 𝐿, we need to cube root both sides of the equation. The cube root of 𝐿 cubed is simply 𝐿 because the cube root is the opposite of cubing, so they cancel each other. And the cube root of 216 is six because six times six times six is equal to 216. So once again, the length of the cube will be six centimeters.