What is the height of a cuboid whose volume is 17𝑥 to the fourth minus six 𝑥 squared 𝑦 centimetres cubed and whose base is a square of side 𝑥 centimetres?
A cuboid, which you might know as a rectangular prism, has six flat faces. And all the angles are right angles. And our cuboid has a base that’s a square. You can imagine that our cuboid looks something like this. The volume of a cuboid or a rectangular prism is found by multiplying its length times its width times its height or, as we sometimes say, the area of the base times the height.
And for us, the base is a square. And that square has a side length of 𝑥 centimetres. The area of the base of this cuboid equals side times side, 𝑥 times 𝑥, which we would call 𝑥 squared centimetres squared.
The volume is 17𝑥 to the fourth minus six 𝑥 squared times 𝑦. We know that 17𝑥 to the fourth minus six 𝑥 squared 𝑦 is equal to the area of the base times the height. And we’ll plug in 𝑥 squared for the area of the base. And now, we need to solve for ℎ, solve for the height.
We can divide the right-hand side of this equation by 𝑥 squared. And if we do that, we’ll need to divide the left-hand side of this equation by 𝑥 squared. 𝑥 squared divided by 𝑥 squared cancels out. And the height of this cuboid has to equal 17𝑥 to the fourth divided by 𝑥 squared minus six 𝑥 squared 𝑦 divided by 𝑥 squared.
For the first term, we have an 𝑥 squared in the denominator and 𝑥 to the fourth in the numerator. The denominator cancels out. Four minus two equals two. The remaining part of that term is 17𝑥 squared. In the second term, we have an 𝑥 squared in the denominator and the 𝑥 squared in the numerator. And they cancel out. The remaining part of that term is six 𝑦. The height of this cuboid is 17𝑥 squared minus six 𝑦. And that’s a measurement of centimetres.
If we wanted to check and see if this was true, we would remember that volume equals area of the base times the height. The base of this cuboid is a square of side length 𝑥. So its area is 𝑥 squared. We would plug in what we found for the height, 17𝑥 squared minus six 𝑦. Distribute the 𝑥 squared. 𝑥 squared times 17𝑥 squared equals 17𝑥 to the fourth. 𝑥 squared times negative six 𝑦 equals negative six 𝑥 squared 𝑦.
And it’s that what we started with. 17𝑥 to the fourth minus six 𝑥 squared 𝑦 is what we started with. And that confirms that we found the correct height.