The city
 
Adelaide, South Australia, a city of a million people, stretches north and south along a coastal plain bounded by the sea in the west and a range of hills in the east. The site was chosen by Colonel William Light in 1836, and the city was named after Queen Adelaide, wife of King William IV. Adelaide is sometimes known as ‘the city of churches’, but could equally well be called ‘the city of parks and gardens’. The central business district is surrounded by a belt of parklands. In the northeast corner of the parklands lie the botanical gardens, and in particular the Bicentennial Conservatory, built in 1988 as one of South Australia’s major contributions to the Australian Bicentennial celebrations.   

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The conservatory

The conservatory, designed by architect Guy Maron, is of particular mathematical interest because the design is based on two identical cones having a common (vertical) base. This design feature leads not only to a striking structure, but also to economies of construction, since the curvature of the surface of the cone is constant.
The surface is splayed at the ends to allow access to the conservatory.




















Properties of the cone

A cone with base radius r and height h has volume V = r2h. We can prove this using calculus, but it is an example of the more general formula for the volume of a pyramid:

V = area of base height.

 Investigate  What formulae give the volumes of the cube? prism? sphere?

The area of the curved surface of a cone is rather easier to find. For if we lay the surface out flat, we obtain a sector of a circle of radius l (like the coloured region illustrated below right).

 Investigate  What is the area of the whole circle? the perimeter of the circle? the length of the curved boundary of the coloured region? What proportion of the circle is shaded?

We see that the area A of the curved surface of a cone of slant height l and base radius r is

 A = l2 = rl. 






















The use of a model

After preparing some initial sketches, the architect will often decide to construct a small scale model to help visualize the building. Such models may be quite simple or very elaborate; in any case it is more economical to make alterations to the model than to the structure itself!

Shown here is a ‘cut-out’ of one of the curved faces of the conservatory which could be used as a component of such a model. It is drawn to a scale of 1:1000. The base of the model is at the left. The triangular flaps fold down inside and are glued together.

 Investigate  What are the relationships between the model and the actual conservatory? How are the length, volume, glass panel areas of the two related?

  Project  Choose some simple structure in your own town (for example a water tower, a dam). Using a suitable scale, make a scale model of it. Discuss how the measurements of the structure relate to the measurements of the model.




















Temperature

An old book contains the information that ideally the temperature in a tropical greenhouse should lie between 68° and 91° Fahrenheit. In this old scale, freezing point is 32° F and boiling point is 212° F.

 Investigate  Find a formula which relates temperatures in the two scales. What would the suggested temperature range be, expressed in degrees Centigrade?

The equation relating the quantities f and c is said to be linear, as it involves no square or higher powers of the variables. If we want to convert a number of temperature measurements from one scale to the other, we can either write a small program for a calculator or computer, or draw a graph.

 Investigate  With suitable scales, draw the straight line graph which relates the two sets of values. Use your graph to convert several temperatures from one scale to the other. Check your results from your formula.

 Link  http://www.environment.sa.govt.au/botanicgardens/conservatory.html