A spherical ball with radius eight centimetres is completely filled with water. The water is then poured into a cylindrical trough of the same radius. The water fills exactly half of the cylinder. Volume of a sphere is equal to four-thirds 𝜋𝑟 cubed, where 𝑟 is the radius. Volume of a cylinder is equal to 𝜋𝑟 squared ℎ, where 𝑟 is the radius and ℎ is the perpendicular height. Work out the length of the cylinder.
To find the length of the cylinder, we’ll need to do two things. First, we’ll need to find the volume of water that could be contained in the spherical ball. We’re then told that when the water is poured into the cylinder, it fills half of that cylinder. We can then double the value of the volume of the sphere to find the total volume of the cylinder. Then we’ll be able to work backwards using the formula for volume of a cylinder to find its length.
The formula for volume of a sphere is given as four-thirds 𝜋𝑟 cubed. We are told that the radius of our sphere is eight centimetres. So we can substitute this value into the formula. And we see that the volume can be calculated by multiplying four-thirds by 𝜋 and then multiplying that by eight cubed.
If we type that into our calculator, we get 2048 over three 𝜋. It’s sensible to leave this answer in terms of 𝜋 for now. This will reduce the risk of making any errors from rounding our answer too early.
The volume of the sphere is 2048 over three 𝜋 centimetres cubed. The water fills exactly half of the cylinder. So now that we know the total amount of water there is, we can double that and it will tell us the volume of the cylinder. That’s two multiplied by 2048 over three 𝜋. The volume of our cylinder then is 4096 over three 𝜋 centimetres cubed.
We now know the volume and the radius of the cylinder. The formula for volume of a cylinder is given to us as 𝜋𝑟 squared ℎ. In fact, this is one we should know by heart. A cylinder is a prism with a circular cross section. And the volume of a prism is found by multiplying the area of its cross section by its height or its length. Since the area of a circle is 𝜋𝑟 squared, we can deduce that the volume of the cylinder is 𝜋𝑟 squared multiplied by height, as given.
Let’s substitute everything we know about our cylinder into the formula for its volume. Its volume is 4096 over three 𝜋, and its radius is eight. We need to solve this equation for ℎ. Currently, ℎ is being multiplied by 𝜋 multiplied by eight squared. To solve for ℎ, we’re going to divide through by 𝜋 multiplied by eight squared.
In fact, we need to be sure that we’re multiplying trough by 𝜋 and eight squared. So let’s put this in brackets. The height of our cylinder is 4096 over three 𝜋 divided by 𝜋 multiplied by eight squared. The two values of 𝜋 end up cancelling each other out. And we’re left with ℎ is being equal to 64 over three centimetres.
Remember, the generic height is actually given as 𝑙, the length of our cylinder. So we can say that 𝑙 is equal to 64 over three or sixty-four thirds centimetres. We could have converted this to a decimal. It’s 21.3 recurring. However, leaving it as a fraction in its simplest form is practical and accurate. 𝑙 is equal to 64 over three centimetres.