Question
Dear Sir/Madam,
I would like to ask a few question. How is the shape of a water tower will affect its stability? Can you please suggest at least 5 examples, such as sphere, cuboids, oval, cylinder, cone and others. Which of those will have maximum volumes, lowest cost of buiding and most stable? Can you please provide the answer with mathematical formula. I am looking forward for your reply. Thank you very much.
Answer
Dear Tan,<BR><BR>I am not sure I can do all the work for you and give you the answer -- it takes some time to look up the various formulas. But perhaps it will help for me to tell you what you need to look up, and you'll be able to find it yourself, even on the Internet.<BR><BR>A water tower has 3 parts: the container that holds the water, the water, and the tower the container is standing on, if any. The container is a "solid shell". See the "shells" mentioned on <A href=http://www.efunda.com/math/solids/IndexSolid.cfm">http://www.efunda.com/math/solids/IndexSolid.cfm</A> for example. The water, for purposes of stability, can be considered as a solid. In many cases, the water will be heavier than the container, so maybe think of the tower as only 2 pieces: a solid shape standing on a supporting tower, or if no tower, standing on the ground.<BR><BR>You asked about stability, volume, and cost of building.<BR><BR>Stability is based on where the center of gravity is. Higher center of gravity is less stable, lower center of gravity is more stable. Center of gravity is also called "centroid". Again see <A href=http://www.efunda.com/math/solids/IndexSolid.cfm">http://www.efunda.com/math/solids/IndexSolid.cfm</A> for centroid formulas. Keep in mind that if your water tower has two pieces, for example a spherical ball of water on top of a narrow pyramid tower, then you'll have to calculate the centroid of the combined object. Find a book on solid geometry and read about calculating centroids.<BR><BR>Maximum volume -- what you mean is "for a given surface area" since that's what determines cost of materials. So look at formulas for volume and divide by formula for surface area -- see which is largest. Without thinking too much, I bet the answer is a sphere. But check yourself.<BR><BR>Lowest cost of building is easy in theory, but harder in fact. In theory, cost of building is cost of materials and materials = surface area of container and volume (or even just height) of tower. But in real life a lot of what determines cost is supply. In other words, whatever kinds of water towers are most built are the cheapest because building companies are set up to build those. Call the water company in your city, ask for "Public Relations", explain your school project, and see if they can offer someone there who can help answer the cost questions. Who knows, you or your teacher might even be able to get the person to come speak to your whole class.<BR><BR>Good luck!
http://www.efunda.com/math/solids/IndexSolid.cfm
centroid<BR>center of gravity<BR>solid geometry<BR>volume<BR>surface area<BR>construction cost estimation
http://vrd.askvrd.org/services/answerschema.xml
