Given that the volume of the smaller cube is eight cubic feet, determine the volume of the larger cube.
First let’s recall how to find the volume of a cube. The volume is equal to 𝑒 cubed, where 𝑒 represents the edge length of the cube. We can see that the edge lengths of these cubes are 𝑥 feet for the smaller cube and three 𝑥 feet for the larger cube.
There are two ways that we could approach this question. The first way is to use the fact that we know the volume of the smaller cube in order to calculate the value of 𝑥. We can then calculate the side length of the larger cube and hence its volume.
As the volume of the smaller cube is eight cubic feet and the edge length is 𝑥 feet, we have the equation 𝑥 cubed is equal to eight. Finding the cube root of each side of this equation gives 𝑥 is equal to the cubed root of eight, which is two.
Now we know the value of 𝑥, we can calculate the edge length of the second cube as it’s equal to three 𝑥. Three 𝑥 is equal to three multiplied by two, which is six. So the edge length of the second cube is six feet.
The volume of the larger cube is therefore equal to its edge length cubed, six cubed, which is 216. Units for this volume are cubic feet. So that’s the first approach that we could take to this question: calculating the value of 𝑥, the edge length of the smaller cube, directly.
The second approach doesn’t actually require us to find the value of 𝑥, but instead uses the relationship between the volumes of the two cubes. We know that the lengths of the two cubes are in the ratio one to three as they are 𝑥 and three 𝑥 feet, respectively. Does this mean that the volumes are also in the ratio one to three?
Volume is calculated by multiplying three dimensions together, each of which are three times larger in the bigger cube compared to the smaller cube. Therefore, the overall volume isn’t three times bigger. It’s three times three times three, or three cubed, times bigger. The volumes of the two cubes are therefore in the ratio one to 27.
We could find the volume of the larger cube by taking the volume of the smaller cube, eight cubic feet, and multiplying it by 27. Eight multiplied by 27 is 216, giving the same answer as we found using the first method. If you’re going to use the second method, then just be very careful.
A really common mistake could be to assume that the volumes are in the same ratio as the lengths and, therefore, that the volume of the larger cube is just three times bigger than the volume of the smaller cube.
This would give a volume of 24 cubic feet, which as we’ve seen is incorrect. The correct volume for the larger cube is 216 cubic feet.