Volume of 3D shapes: Comparing, rates of flow

In a nutshell

You can apply your knowledge of the volumes of 3D shapes to approach more complicated problems that involve comparing volumes and rates of flow.

Comparing volumes

You may be asked to calculate the volumes of two 3D shapes and compare them. To solve these types of questions, always use the volume formulae and compare after working out the volume for each shape.

Example 1

A hemisphere of radius $R$ has the same volume as a sphere of radius $r$ . What is the value of $\dfrac{R}{r}$ ?

First, find the volumes of the two shapes.

The volume of the sphere is given by the formula:

$V_{sphere}=\dfrac{4}{3}\pi r^3$

A hemisphere is a half of a sphere. This means that the volume of the hemisphere is:

$V_{hemisphere}=\dfrac{1}{2}\times\dfrac{4}{3}\pi R^3=\dfrac{2}{3}\pi R^3$

It is given that the two volumes are equal. So, set these two expressions equal to one another:

$V_{sphere}=V_{hemisphere}$

$\dfrac{4}{3}\pi r^3=\dfrac{2}{3}\pi R^3$

$4r^3=2R^3$

$\dfrac{R^3}{r^3}=2$

$\bigg( \dfrac{R}{r} \bigg)^3=2$

$\underline{\dfrac{R}{r}=\sqrt[3]{2}}$

Rates of flow

Rates of flow questions involve a 3D shape that has liquid flowing in or out of the shape at a constant rate. To appoach these questions, use the formula:

$\text{rate of flow}=\dfrac{\text{volume}}{\text{time}}$

Example 2

A cubic tank fills with water at a rate of $15cm^3/s$ . It takes $30$ minutes for the tank to fill completely. What is the length of one side of the cube?

First, make sure all the units are consistent. Convert $30$ minutes to seconds:

$30\times60=1800$ seconds

Now, rearrange the formula for the volume, and substitute in the values:

$\text{rate of flow}=\dfrac{\text{volume}}{\text{time}}\Rightarrow \text{volume}=\text{time}\times\text{rate of flow}$

$V=1800\times15=27000$

$V=27000cm^3$

Use the formula for the volume of a cube to work out the length of one side:

$V=x^3=27000$

$x=\sqrt[3]{27000}=30cm$

The length of one side is $\underline{30cm}$ .