A cylinder has a volume of 54 𝜋 centimetres cubed. Given that its height is equal to the radius of its base, find its height.
So in this question, we’re told that the volume of the cylinder is 54 𝜋 centimetres cubed. We aren’t told either the height or the radius of the cylinder, but we are told the important piece of information that they are equal to each other. Let’s think about how to approach this problem.
As we’re told the volume of the cylinder, let’s begin by writing down the formula to calculate the volume of a cylinder. It’s 𝜋𝑟 squared ℎ, where 𝑟 represents the radius and ℎ represents the height. For our cylinder which has a volume of 54 𝜋, this means that 𝜋𝑟 squared ℎ must be equal to 54 𝜋. As there’s a factor of 𝜋 on both sides of this equation, they cancel each other out directly. This leaves us with 𝑟 squared ℎ is equal to 54.
Now, this is one equation with two letters, which means we wouldn’t usually be able to solve it, but remember we know that the radius and the height are equal to each other. As we’re looking to find the height of a cylinder, this means that we can replace the letter 𝑟 in this equation with the letter ℎ as 𝑟 is equal to ℎ. So we have that ℎ squared multiplied by ℎ is equal to 54. And now, this equation has only one unknown letter. ℎ squared multiplied by ℎ is ℎ cubed, so we have ℎ cubed is equal to 54.
To find the value of ℎ, we need to take the cube root of each side. We have that ℎ is equal to the cube root of 54. Now, this can be simplified slightly if you recall that 54 is equal to 27 multiplied by two and 27 is a cubed number. We can separate this cube root into the cube root of 27 multiplied by the cube root of two.
Now, remember 27 is a cubed number; it’s equal to three multiplied by three multiplied by three, which means that the first term here just simplifies to three. Therefore, this simplifies to three multiplied by the cube root of two. Therefore, the height of the cylinder and incidentally also the radius is three multiplied by the cube root of two and the units for this are centimetres.