### In a Spin

What is the volume of the solid formed by rotating this right angled triangle about the hypotenuse?

### Conical Bottle

A right circular cone is filled with liquid to a depth of half its vertical height. The cone is inverted. How high up the vertical height of the cone will the liquid rise?

### Peeling the Apple or the Cone That Lost Its Head

How much peel does an apple have?

# Fill Me up Too

### Why do this problem?

This problem follows on from Fill Me Up, and gives students the opportunity to use volume scale factors of enlargement to work out the relationship between the volume and the height of a cone.

### Possible approach

Perhaps start by asking students to sketch the graphs from the problem Fill Me Up. Here is a worksheet showing the containers.

"Imagine we wanted to plot the graphs accurately by working out the equations linking height to volume. Some parts of the containers will be easier to work out than others - which will be easiest? Which will be hardest?"

Take time to discuss students' ideas, relating it back to the graphs sketched in the first problem.

"Let's try to analyse how the height changes as the Pint Glass is filled."
"The Pint Glass can be thought of as part of a cone (a frustum), so I'd like you to consider a cone filling with water first."
Give students this worksheet to work on in groups of 3 or 4. These roles may be useful for students who are not used to working collaboratively on a problem. Make it clear that your expectation is for all students in the group to be able to explain their thinking clearly and that anyone might be chosen to present the group's conclusions at the end of the lesson.

Finally, allow time at the end of the lesson (or two lessons) for groups to present their thinking to the rest of the class.

### Key questions

What happens to the volume of a cone when I enlarge it by a scale factor of 2, 3, 4, 5... k?
If the volume of water is $10$cm$^3$ when the height of the water is $1$cm, what will the volume be when the height is $2, 3, 4...x$cm?
How could this be represented graphically?

### Possible extension

There are two extension tasks suggested in the problem: analysing the inverted cone is a reasonably straightforward extension, but analysing the spherical flask is much much more challenging.

Immersion and Brimful both offer extension possibilities for considering functional relationships relating to volume.

### Possible support

Growing Rectangles offers a good introduction to proportional relationships between length, area and volume.

The NRICH Project aims to enrich the mathematical experiences of all learners. To support this aim, members of the NRICH team work in a wide range of capacities, including providing professional development for teachers wishing to embed rich mathematical tasks into everyday classroom practice. More information on many of our other activities can be found here.
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