Qqq..cubed

It is known that the area of the largest equilateral triangular section of a cube is 140sq cm. What is the side length of the cube? The distances between the centres of two adjacent faces of another cube is 8cms. What is the side length of this cube? Another cube has an edge length of 12cm. At each vertex a tetrahedron with three mutually perpendicular edges of length 4cm is sliced away. What is the surface area and volume of the remaining solid?

Concrete Calculation

The builders have dug a hole in the ground to be filled with concrete for the foundations of our garage. How many cubic metres of ready-mix concrete should the builders order to fill this hole to make the concrete raft for the foundations?

In a Spin

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

Immersion

Why do this problem?

This problem provides an opportunity to think about what the shape of a graph means, by relating the graph to the situation which created it. Students need to reason which graph matches with which solid, and clearly communicate their ideas. Along the way, there is the chance to revisit formulas for working out the volumes of solids.

Possible approach

Start by asking students to sketch the four solids described in the problem:
1) A sphere of radius $1$cm
2) A solid cylinder with height $\frac{4}{3}$cm and radius $1$cm
3) A solid circular cone with base radius $1$cm and height $4$cm.
4) A solid cylinder of height $\frac{4}{9}$cm with a hole drilled through it, leaving an annular (ring-shaped) cross-section with internal radius $1$cm and external radius $2$cm.

Then for each of the solids, students could work out the volume - this is a good opportunity to discuss the advantages of leaving answers in terms of $\pi$.

Now hand out this worksheet with the five graphs on. Challenge students to label the axes and the five graphs, annotating each curve along the way to show key points where interesting things are happening, and how they relate to the shape of the solid.

Finally, bring the class together to discuss their answers and how they pieced together which graph is which.

Key questions

What does a straight line section of graph tell you about the cross section of the solid being immersed?
What about a curve which gets steeper and steeper? Or one which flattens off gradually?
Are there any symmetrical features of the solids which help to identify the correct lines?

Possible extension

See extension task at the end of the problem.

Possible support

Maths Filler offers an opportunity to work with graphs showing changing volumes based on simple surface areas and graphs made up of straight lines rather than curves.