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 rR?
First, find the volumes of the two shapes.
The volume of the sphere is given by the formula:
Vsphere=34πr3
A hemisphere is a half of a sphere. This means that the volume of the hemisphere is:
Vhemisphere=21×34πR3=32πR3
It is given that the two volumes are equal. So, set these two expressions equal to one another:
Vsphere=Vhemisphere
34πr3=32πR3
4r3=2R3
r3R3=2
(rR)3=2
rR=32
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:
rate of flow=timevolume
Example 2
A cubic tank fills with water at a rate of 15cm3/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:
30×60=1800 seconds
Now, rearrange the formula for the volume, and substitute in the values:
rate of flow=timevolume⇒volume=time×rate of flow
V=1800×15=27000
V=27000cm3
Use the formula for the volume of a cube to work out the length of one side:
First, make sure all the units are consistent with each other. Then, rearrange the rate of flow equation for the desired quantity and substitute in the known values.
What is the formula for rate of flow?
The formula for rate of flow is rate of flow = volume/time.
How do you solve problems about comparing volumes?
Work out the volume of each shape using their respective formula. Then, compare the volumes.